From 201d649f7c799d24a5eb3e45848ce1f2681f3ab3 Mon Sep 17 00:00:00 2001
From: rockysaikia730 <67681828+rockysaikia730@users.noreply.github.com>
Date: Wed, 7 Oct 2020 21:45:21 +0530
Subject: [PATCH 01/18] Property of adjoint
---
adjoint.ipynb | 76 +++++++++++++++++++++++++++++++++++++++++++++++++++
1 file changed, 76 insertions(+)
create mode 100644 adjoint.ipynb
diff --git a/adjoint.ipynb b/adjoint.ipynb
new file mode 100644
index 0000000..18dfd13
--- /dev/null
+++ b/adjoint.ipynb
@@ -0,0 +1,76 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "**1.**\n",
+ "### $A.adj(A)= adj(A).A = |A|I$\n",
+ "\n",
+ "#### Let A be a square matrix of order n and C represent the cofactor matrix of A and adj(A) represent the adjoint of matrix A .\n",
+ "\n",
+ "$$(adj(A))_{ij} = [C_{ij}]^T = C_{ji}$$\n",
+ "
\n",
+ "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.(adj(A))_{rj}$$\n",
+ "
\n",
+ "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr}$$\n",
+ "
\n",
+ "#### We Know,\n",
+ "\n",
+ "##### Sum of products of elements with their corresponding cofactor is equal to the determinant of the matrix.\n",
+ "
\n",
+ "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr} = \\left\\{\\begin{array}\\\\ \n",
+ "|A| & \\mbox{if } \\ i=j\\\\ \n",
+ "0 & \\mbox{if } \\ i \\neq j\n",
+ "\\end{array}\\right.$$\n",
+ "\n",
+ "\n",
+ "#### So,\n",
+ "
\n",
+ "$$ A.adj(A) = \\begin{bmatrix} |A| & 0 & 0 & . & . & 0\n",
+ " \\\\ 0 & |A| & 0 & . & . & 0\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ 0 & 0 & . & . & . & |A|\n",
+ " \\end{bmatrix}_{n\\times n}$$\n",
+ " \n",
+ "
\n",
+ "$$ A.adj(A) = |A|.\\begin{bmatrix} 1 & 0 & 0 & . & . & 0\n",
+ " \\\\ 0 & 1 & 0 & . & . & 0\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ 0 & 0 & . & . & . & 1\n",
+ " \\end{bmatrix}_{n\\times n}$$\n",
+ " \n",
+ "
\n",
+ "$$A.adj(A) = |A|.I_{n}$$\n",
+ "\n",
+ "\n",
+ "##### Similarly we can prove adj(A).A = |A|.I\n",
+ "\n",
+ "\n"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.7.3"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
From ba6f7ad4d67433524f35f22dd9347227a7df75b2 Mon Sep 17 00:00:00 2001
From: rockysaikia730 <67681828+rockysaikia730@users.noreply.github.com>
Date: Wed, 7 Oct 2020 21:46:42 +0530
Subject: [PATCH 02/18] Delete adjoint.ipynb
---
adjoint.ipynb | 76 ---------------------------------------------------
1 file changed, 76 deletions(-)
delete mode 100644 adjoint.ipynb
diff --git a/adjoint.ipynb b/adjoint.ipynb
deleted file mode 100644
index 18dfd13..0000000
--- a/adjoint.ipynb
+++ /dev/null
@@ -1,76 +0,0 @@
-{
- "cells": [
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "**1.**\n",
- "### $A.adj(A)= adj(A).A = |A|I$\n",
- "\n",
- "#### Let A be a square matrix of order n and C represent the cofactor matrix of A and adj(A) represent the adjoint of matrix A .\n",
- "\n",
- "$$(adj(A))_{ij} = [C_{ij}]^T = C_{ji}$$\n",
- "
\n",
- "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.(adj(A))_{rj}$$\n",
- "
\n",
- "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr}$$\n",
- "
\n",
- "#### We Know,\n",
- "\n",
- "##### Sum of products of elements with their corresponding cofactor is equal to the determinant of the matrix.\n",
- "
\n",
- "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr} = \\left\\{\\begin{array}\\\\ \n",
- "|A| & \\mbox{if } \\ i=j\\\\ \n",
- "0 & \\mbox{if } \\ i \\neq j\n",
- "\\end{array}\\right.$$\n",
- "\n",
- "\n",
- "#### So,\n",
- "
\n",
- "$$ A.adj(A) = \\begin{bmatrix} |A| & 0 & 0 & . & . & 0\n",
- " \\\\ 0 & |A| & 0 & . & . & 0\n",
- " \\\\ . & . & . & . & . & .\n",
- " \\\\ . & . & . & . & . & .\n",
- " \\\\ 0 & 0 & . & . & . & |A|\n",
- " \\end{bmatrix}_{n\\times n}$$\n",
- " \n",
- "
\n",
- "$$ A.adj(A) = |A|.\\begin{bmatrix} 1 & 0 & 0 & . & . & 0\n",
- " \\\\ 0 & 1 & 0 & . & . & 0\n",
- " \\\\ . & . & . & . & . & .\n",
- " \\\\ . & . & . & . & . & .\n",
- " \\\\ 0 & 0 & . & . & . & 1\n",
- " \\end{bmatrix}_{n\\times n}$$\n",
- " \n",
- "
\n",
- "$$A.adj(A) = |A|.I_{n}$$\n",
- "\n",
- "\n",
- "##### Similarly we can prove adj(A).A = |A|.I\n",
- "\n",
- "\n"
- ]
- }
- ],
- "metadata": {
- "kernelspec": {
- "display_name": "Python 3",
- "language": "python",
- "name": "python3"
- },
- "language_info": {
- "codemirror_mode": {
- "name": "ipython",
- "version": 3
- },
- "file_extension": ".py",
- "mimetype": "text/x-python",
- "name": "python",
- "nbconvert_exporter": "python",
- "pygments_lexer": "ipython3",
- "version": "3.7.3"
- }
- },
- "nbformat": 4,
- "nbformat_minor": 2
-}
From 2142fb110b992dc829c2a80a36d3f59f9de63a7f Mon Sep 17 00:00:00 2001
From: rockysaikia730 <67681828+rockysaikia730@users.noreply.github.com>
Date: Wed, 7 Oct 2020 21:47:37 +0530
Subject: [PATCH 03/18] Property of Adjoint
---
Basics/adjoint.ipynb | 76 ++++++++++++++++++++++++++++++++++++++++++++
1 file changed, 76 insertions(+)
create mode 100644 Basics/adjoint.ipynb
diff --git a/Basics/adjoint.ipynb b/Basics/adjoint.ipynb
new file mode 100644
index 0000000..18dfd13
--- /dev/null
+++ b/Basics/adjoint.ipynb
@@ -0,0 +1,76 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "**1.**\n",
+ "### $A.adj(A)= adj(A).A = |A|I$\n",
+ "\n",
+ "#### Let A be a square matrix of order n and C represent the cofactor matrix of A and adj(A) represent the adjoint of matrix A .\n",
+ "\n",
+ "$$(adj(A))_{ij} = [C_{ij}]^T = C_{ji}$$\n",
+ "
\n",
+ "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.(adj(A))_{rj}$$\n",
+ "
\n",
+ "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr}$$\n",
+ "
\n",
+ "#### We Know,\n",
+ "\n",
+ "##### Sum of products of elements with their corresponding cofactor is equal to the determinant of the matrix.\n",
+ "
\n",
+ "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr} = \\left\\{\\begin{array}\\\\ \n",
+ "|A| & \\mbox{if } \\ i=j\\\\ \n",
+ "0 & \\mbox{if } \\ i \\neq j\n",
+ "\\end{array}\\right.$$\n",
+ "\n",
+ "\n",
+ "#### So,\n",
+ "
\n",
+ "$$ A.adj(A) = \\begin{bmatrix} |A| & 0 & 0 & . & . & 0\n",
+ " \\\\ 0 & |A| & 0 & . & . & 0\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ 0 & 0 & . & . & . & |A|\n",
+ " \\end{bmatrix}_{n\\times n}$$\n",
+ " \n",
+ "
\n",
+ "$$ A.adj(A) = |A|.\\begin{bmatrix} 1 & 0 & 0 & . & . & 0\n",
+ " \\\\ 0 & 1 & 0 & . & . & 0\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ 0 & 0 & . & . & . & 1\n",
+ " \\end{bmatrix}_{n\\times n}$$\n",
+ " \n",
+ "
\n",
+ "$$A.adj(A) = |A|.I_{n}$$\n",
+ "\n",
+ "\n",
+ "##### Similarly we can prove adj(A).A = |A|.I\n",
+ "\n",
+ "\n"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.7.3"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
From 5aee509bcba0392d6378907238ecc5d32c3f8004 Mon Sep 17 00:00:00 2001
From: rockysaikia730 <67681828+rockysaikia730@users.noreply.github.com>
Date: Wed, 7 Oct 2020 21:57:57 +0530
Subject: [PATCH 04/18] property of adjoint
---
Basics/adjoint.md | 47 +++++++++++++++++++++++++++++++++++++++++++++++
1 file changed, 47 insertions(+)
create mode 100644 Basics/adjoint.md
diff --git a/Basics/adjoint.md b/Basics/adjoint.md
new file mode 100644
index 0000000..a40bb0b
--- /dev/null
+++ b/Basics/adjoint.md
@@ -0,0 +1,47 @@
+
+**1.**
+### $A.adj(A)= adj(A).A = |A|I$
+
+#### Let A be a square matrix of order n and C represent the cofactor matrix of A and adj(A) represent the adjoint of matrix A .
+
+$$(adj(A))_{ij} = [C_{ij}]^T = C_{ji}$$
+
+$$(A.adj(A))_{ij} = \sum_{r=1}^{r=n} a_{ir}.(adj(A))_{rj}$$
+
+$$(A.adj(A))_{ij} = \sum_{r=1}^{r=n} a_{ir}.C_{jr}$$
+
+#### We Know,
+
+##### Sum of products of elements with their corresponding cofactor is equal to the determinant of the matrix.
+
+$$(A.adj(A))_{ij} = \sum_{r=1}^{r=n} a_{ir}.C_{jr} = \left\{\begin{array}\\
+|A| & \mbox{if } \ i=j\\
+0 & \mbox{if } \ i \neq j
+\end{array}\right.$$
+
+
+#### So,
+
+$$ A.adj(A) = \begin{bmatrix} |A| & 0 & 0 & . & . & 0
+ \\ 0 & |A| & 0 & . & . & 0
+ \\ . & . & . & . & . & .
+ \\ . & . & . & . & . & .
+ \\ 0 & 0 & . & . & . & |A|
+ \end{bmatrix}_{n\times n}$$
+
+
+$$ A.adj(A) = |A|.\begin{bmatrix} 1 & 0 & 0 & . & . & 0
+ \\ 0 & 1 & 0 & . & . & 0
+ \\ . & . & . & . & . & .
+ \\ . & . & . & . & . & .
+ \\ 0 & 0 & . & . & . & 1
+ \end{bmatrix}_{n\times n}$$
+
+
+$$A.adj(A) = |A|.I_{n}$$
+
+
+##### Similarly we can prove adj(A).A = |A|.I
+
+
+
From 95da48a6fefb6d0dc1793daa5beeb12236f32d8b Mon Sep 17 00:00:00 2001
From: rockysaikia730 <67681828+rockysaikia730@users.noreply.github.com>
Date: Wed, 7 Oct 2020 21:58:23 +0530
Subject: [PATCH 05/18] Delete adjoint.md
---
Basics/adjoint.md | 47 -----------------------------------------------
1 file changed, 47 deletions(-)
delete mode 100644 Basics/adjoint.md
diff --git a/Basics/adjoint.md b/Basics/adjoint.md
deleted file mode 100644
index a40bb0b..0000000
--- a/Basics/adjoint.md
+++ /dev/null
@@ -1,47 +0,0 @@
-
-**1.**
-### $A.adj(A)= adj(A).A = |A|I$
-
-#### Let A be a square matrix of order n and C represent the cofactor matrix of A and adj(A) represent the adjoint of matrix A .
-
-$$(adj(A))_{ij} = [C_{ij}]^T = C_{ji}$$
-
-$$(A.adj(A))_{ij} = \sum_{r=1}^{r=n} a_{ir}.(adj(A))_{rj}$$
-
-$$(A.adj(A))_{ij} = \sum_{r=1}^{r=n} a_{ir}.C_{jr}$$
-
-#### We Know,
-
-##### Sum of products of elements with their corresponding cofactor is equal to the determinant of the matrix.
-
-$$(A.adj(A))_{ij} = \sum_{r=1}^{r=n} a_{ir}.C_{jr} = \left\{\begin{array}\\
-|A| & \mbox{if } \ i=j\\
-0 & \mbox{if } \ i \neq j
-\end{array}\right.$$
-
-
-#### So,
-
-$$ A.adj(A) = \begin{bmatrix} |A| & 0 & 0 & . & . & 0
- \\ 0 & |A| & 0 & . & . & 0
- \\ . & . & . & . & . & .
- \\ . & . & . & . & . & .
- \\ 0 & 0 & . & . & . & |A|
- \end{bmatrix}_{n\times n}$$
-
-
-$$ A.adj(A) = |A|.\begin{bmatrix} 1 & 0 & 0 & . & . & 0
- \\ 0 & 1 & 0 & . & . & 0
- \\ . & . & . & . & . & .
- \\ . & . & . & . & . & .
- \\ 0 & 0 & . & . & . & 1
- \end{bmatrix}_{n\times n}$$
-
-
-$$A.adj(A) = |A|.I_{n}$$
-
-
-##### Similarly we can prove adj(A).A = |A|.I
-
-
-
From 2bb329cc0b90cb5e3cb10f67acba3cdc6e4953eb Mon Sep 17 00:00:00 2001
From: rockysaikia730 <67681828+rockysaikia730@users.noreply.github.com>
Date: Wed, 7 Oct 2020 21:59:00 +0530
Subject: [PATCH 06/18] Delete adjoint.ipynb
---
Basics/adjoint.ipynb | 76 --------------------------------------------
1 file changed, 76 deletions(-)
delete mode 100644 Basics/adjoint.ipynb
diff --git a/Basics/adjoint.ipynb b/Basics/adjoint.ipynb
deleted file mode 100644
index 18dfd13..0000000
--- a/Basics/adjoint.ipynb
+++ /dev/null
@@ -1,76 +0,0 @@
-{
- "cells": [
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "**1.**\n",
- "### $A.adj(A)= adj(A).A = |A|I$\n",
- "\n",
- "#### Let A be a square matrix of order n and C represent the cofactor matrix of A and adj(A) represent the adjoint of matrix A .\n",
- "\n",
- "$$(adj(A))_{ij} = [C_{ij}]^T = C_{ji}$$\n",
- "
\n",
- "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.(adj(A))_{rj}$$\n",
- "
\n",
- "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr}$$\n",
- "
\n",
- "#### We Know,\n",
- "\n",
- "##### Sum of products of elements with their corresponding cofactor is equal to the determinant of the matrix.\n",
- "
\n",
- "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr} = \\left\\{\\begin{array}\\\\ \n",
- "|A| & \\mbox{if } \\ i=j\\\\ \n",
- "0 & \\mbox{if } \\ i \\neq j\n",
- "\\end{array}\\right.$$\n",
- "\n",
- "\n",
- "#### So,\n",
- "
\n",
- "$$ A.adj(A) = \\begin{bmatrix} |A| & 0 & 0 & . & . & 0\n",
- " \\\\ 0 & |A| & 0 & . & . & 0\n",
- " \\\\ . & . & . & . & . & .\n",
- " \\\\ . & . & . & . & . & .\n",
- " \\\\ 0 & 0 & . & . & . & |A|\n",
- " \\end{bmatrix}_{n\\times n}$$\n",
- " \n",
- "
\n",
- "$$ A.adj(A) = |A|.\\begin{bmatrix} 1 & 0 & 0 & . & . & 0\n",
- " \\\\ 0 & 1 & 0 & . & . & 0\n",
- " \\\\ . & . & . & . & . & .\n",
- " \\\\ . & . & . & . & . & .\n",
- " \\\\ 0 & 0 & . & . & . & 1\n",
- " \\end{bmatrix}_{n\\times n}$$\n",
- " \n",
- "
\n",
- "$$A.adj(A) = |A|.I_{n}$$\n",
- "\n",
- "\n",
- "##### Similarly we can prove adj(A).A = |A|.I\n",
- "\n",
- "\n"
- ]
- }
- ],
- "metadata": {
- "kernelspec": {
- "display_name": "Python 3",
- "language": "python",
- "name": "python3"
- },
- "language_info": {
- "codemirror_mode": {
- "name": "ipython",
- "version": 3
- },
- "file_extension": ".py",
- "mimetype": "text/x-python",
- "name": "python",
- "nbconvert_exporter": "python",
- "pygments_lexer": "ipython3",
- "version": "3.7.3"
- }
- },
- "nbformat": 4,
- "nbformat_minor": 2
-}
From faaf507e17e057776aa1bc48cbd77170e7013895 Mon Sep 17 00:00:00 2001
From: rockysaikia730 <67681828+rockysaikia730@users.noreply.github.com>
Date: Wed, 7 Oct 2020 22:01:57 +0530
Subject: [PATCH 07/18] Property of Adjoint
---
Basics/adjoint.ipynb | 83 ++++++++++++++++++++++++++++++++++++++++++++
1 file changed, 83 insertions(+)
create mode 100644 Basics/adjoint.ipynb
diff --git a/Basics/adjoint.ipynb b/Basics/adjoint.ipynb
new file mode 100644
index 0000000..3e861dd
--- /dev/null
+++ b/Basics/adjoint.ipynb
@@ -0,0 +1,83 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "**1.**\n",
+ "### $A.adj(A)= adj(A).A = |A|I$\n",
+ "\n",
+ "#### Let A be a square matrix of order n and C represent the cofactor matrix of A and adj(A) represent the adjoint of matrix A .\n",
+ "\n",
+ "$$(adj(A))_{ij} = [C_{ij}]^T = C_{ji}$$\n",
+ "
\n",
+ "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.(adj(A))_{rj}$$\n",
+ "
\n",
+ "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr}$$\n",
+ "
\n",
+ "#### We Know,\n",
+ "\n",
+ "##### Sum of products of elements with their corresponding cofactor is equal to the determinant of the matrix.\n",
+ "\n",
+ "\n",
+ "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr} = \\left\\{\\begin{array}\\\\ |A| & \\mbox{if } \\ i=j \n",
+ " \\\\ 0 & \\mbox{if } \\ i \\neq j\n",
+ "\\end{array}\\right.$$\n",
+ "\n",
+ "\n",
+ "#### So,\n",
+ "
\n",
+ "$$ A.adj(A) = \\begin{bmatrix} |A| & 0 & 0 & . & . & 0\n",
+ " \\\\ 0 & |A| & 0 & . & . & 0\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ 0 & 0 & . & . & . & |A|\n",
+ " \\end{bmatrix}_{n\\times n}$$\n",
+ " \n",
+ "
\n",
+ "$$ A.adj(A) = |A|.\\begin{bmatrix} 1 & 0 & 0 & . & . & 0\n",
+ " \\\\ 0 & 1 & 0 & . & . & 0\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ 0 & 0 & . & . & . & 1\n",
+ " \\end{bmatrix}_{n\\times n}$$\n",
+ " \n",
+ "
\n",
+ "$$A.adj(A) = |A|.I_{n}$$\n",
+ "\n",
+ "\n",
+ "##### Similarly we can prove adj(A).A = |A|.I\n",
+ "\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.7.3"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
From b67ed7f31a890a2e45317624c803ef3040a4f9de Mon Sep 17 00:00:00 2001
From: rockysaikia730 <67681828+rockysaikia730@users.noreply.github.com>
Date: Wed, 7 Oct 2020 22:05:47 +0530
Subject: [PATCH 08/18] Delete adjoint.ipynb
---
Basics/adjoint.ipynb | 83 --------------------------------------------
1 file changed, 83 deletions(-)
delete mode 100644 Basics/adjoint.ipynb
diff --git a/Basics/adjoint.ipynb b/Basics/adjoint.ipynb
deleted file mode 100644
index 3e861dd..0000000
--- a/Basics/adjoint.ipynb
+++ /dev/null
@@ -1,83 +0,0 @@
-{
- "cells": [
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "**1.**\n",
- "### $A.adj(A)= adj(A).A = |A|I$\n",
- "\n",
- "#### Let A be a square matrix of order n and C represent the cofactor matrix of A and adj(A) represent the adjoint of matrix A .\n",
- "\n",
- "$$(adj(A))_{ij} = [C_{ij}]^T = C_{ji}$$\n",
- "
\n",
- "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.(adj(A))_{rj}$$\n",
- "
\n",
- "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr}$$\n",
- "
\n",
- "#### We Know,\n",
- "\n",
- "##### Sum of products of elements with their corresponding cofactor is equal to the determinant of the matrix.\n",
- "\n",
- "\n",
- "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr} = \\left\\{\\begin{array}\\\\ |A| & \\mbox{if } \\ i=j \n",
- " \\\\ 0 & \\mbox{if } \\ i \\neq j\n",
- "\\end{array}\\right.$$\n",
- "\n",
- "\n",
- "#### So,\n",
- "
\n",
- "$$ A.adj(A) = \\begin{bmatrix} |A| & 0 & 0 & . & . & 0\n",
- " \\\\ 0 & |A| & 0 & . & . & 0\n",
- " \\\\ . & . & . & . & . & .\n",
- " \\\\ . & . & . & . & . & .\n",
- " \\\\ 0 & 0 & . & . & . & |A|\n",
- " \\end{bmatrix}_{n\\times n}$$\n",
- " \n",
- "
\n",
- "$$ A.adj(A) = |A|.\\begin{bmatrix} 1 & 0 & 0 & . & . & 0\n",
- " \\\\ 0 & 1 & 0 & . & . & 0\n",
- " \\\\ . & . & . & . & . & .\n",
- " \\\\ . & . & . & . & . & .\n",
- " \\\\ 0 & 0 & . & . & . & 1\n",
- " \\end{bmatrix}_{n\\times n}$$\n",
- " \n",
- "
\n",
- "$$A.adj(A) = |A|.I_{n}$$\n",
- "\n",
- "\n",
- "##### Similarly we can prove adj(A).A = |A|.I\n",
- "\n",
- "\n"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": []
- }
- ],
- "metadata": {
- "kernelspec": {
- "display_name": "Python 3",
- "language": "python",
- "name": "python3"
- },
- "language_info": {
- "codemirror_mode": {
- "name": "ipython",
- "version": 3
- },
- "file_extension": ".py",
- "mimetype": "text/x-python",
- "name": "python",
- "nbconvert_exporter": "python",
- "pygments_lexer": "ipython3",
- "version": "3.7.3"
- }
- },
- "nbformat": 4,
- "nbformat_minor": 2
-}
From 80705f27da86e9f4087ee58c4ab2ee4454946510 Mon Sep 17 00:00:00 2001
From: rockysaikia730 <67681828+rockysaikia730@users.noreply.github.com>
Date: Wed, 7 Oct 2020 22:06:20 +0530
Subject: [PATCH 09/18] Property of adjoint
---
Basics/adjoint.ipynb | 84 ++++++++++++++++++++++++++++++++++++++++++++
1 file changed, 84 insertions(+)
create mode 100644 Basics/adjoint.ipynb
diff --git a/Basics/adjoint.ipynb b/Basics/adjoint.ipynb
new file mode 100644
index 0000000..b01a6d6
--- /dev/null
+++ b/Basics/adjoint.ipynb
@@ -0,0 +1,84 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "**1.**\n",
+ "### $A.adj(A)= adj(A).A = |A|I$\n",
+ "\n",
+ "#### Let A be a square matrix of order n and C represent the cofactor matrix of A and adj(A) represent the adjoint of matrix A .\n",
+ "\n",
+ "$$(adj(A))_{ij} = [C_{ij}]^T = C_{ji}$$\n",
+ "
\n",
+ "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.(adj(A))_{rj}$$\n",
+ "
\n",
+ "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr}$$\n",
+ "
\n",
+ "#### We Know,\n",
+ "\n",
+ "##### Sum of products of elements with their corresponding cofactor is equal to the determinant of the matrix.\n",
+ "
\n",
+ "\n",
+ "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr} = \\left\\{\\begin{array}\\\\ |A| & \\mbox{if } \\ i=j \n",
+ " \\\\ 0 & \\mbox{if } \\ i \\neq j\n",
+ "\\end{array}\\right.$$\n",
+ "\n",
+ "
\n",
+ "\n",
+ "#### So,\n",
+ "
\n",
+ "$$ A.adj(A) = \\begin{bmatrix} |A| & 0 & 0 & . & . & 0\n",
+ " \\\\ 0 & |A| & 0 & . & . & 0\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ 0 & 0 & . & . & . & |A|\n",
+ " \\end{bmatrix}_{n\\times n}$$\n",
+ " \n",
+ "
\n",
+ "$$ A.adj(A) = |A|.\\begin{bmatrix} 1 & 0 & 0 & . & . & 0\n",
+ " \\\\ 0 & 1 & 0 & . & . & 0\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ 0 & 0 & . & . & . & 1\n",
+ " \\end{bmatrix}_{n\\times n}$$\n",
+ " \n",
+ "
\n",
+ "$$A.adj(A) = |A|.I_{n}$$\n",
+ "\n",
+ "\n",
+ "##### Similarly we can prove adj(A).A = |A|.I\n",
+ "\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.7.3"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
From 52ff3fee0103cc4671978cb9c8e5eb3969762bae Mon Sep 17 00:00:00 2001
From: rockysaikia730 <67681828+rockysaikia730@users.noreply.github.com>
Date: Wed, 7 Oct 2020 22:13:37 +0530
Subject: [PATCH 10/18] Update adjoint.ipynb
---
Basics/adjoint.ipynb | 4 ++--
1 file changed, 2 insertions(+), 2 deletions(-)
diff --git a/Basics/adjoint.ipynb b/Basics/adjoint.ipynb
index b01a6d6..b1282c5 100644
--- a/Basics/adjoint.ipynb
+++ b/Basics/adjoint.ipynb
@@ -20,8 +20,8 @@
"##### Sum of products of elements with their corresponding cofactor is equal to the determinant of the matrix.\n",
"
\n",
"\n",
- "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr} = \\left\\{\\begin{array}\\\\ |A| & \\mbox{if } \\ i=j \n",
- " \\\\ 0 & \\mbox{if } \\ i \\neq j\n",
+ "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr} = \\left\\{\\begin{array}\\\\ |A| & \\mbox{if} \\ i=j \n",
+ " \\\\ 0 & \\mbox{if} \\ i \\neq j\n",
"\\end{array}\\right.$$\n",
"\n",
"
\n",
From db8781e243e1e04bcde5ceaae6cf19aaed249ce3 Mon Sep 17 00:00:00 2001
From: rockysaikia730 <67681828+rockysaikia730@users.noreply.github.com>
Date: Wed, 7 Oct 2020 22:16:33 +0530
Subject: [PATCH 11/18] Update adjoint.ipynb
---
Basics/adjoint.ipynb | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
diff --git a/Basics/adjoint.ipynb b/Basics/adjoint.ipynb
index b1282c5..1b7bd03 100644
--- a/Basics/adjoint.ipynb
+++ b/Basics/adjoint.ipynb
@@ -22,7 +22,7 @@
"\n",
"$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr} = \\left\\{\\begin{array}\\\\ |A| & \\mbox{if} \\ i=j \n",
" \\\\ 0 & \\mbox{if} \\ i \\neq j\n",
- "\\end{array}\\right.$$\n",
+ "\\end{array}\\right$$\n",
"\n",
"
\n",
"\n",
From 9e8dbe32a850c6f4eabe8ef63066e6b4f7bfc349 Mon Sep 17 00:00:00 2001
From: rockysaikia730 <67681828+rockysaikia730@users.noreply.github.com>
Date: Wed, 7 Oct 2020 22:21:24 +0530
Subject: [PATCH 12/18] Delete adjoint.ipynb
---
Basics/adjoint.ipynb | 84 --------------------------------------------
1 file changed, 84 deletions(-)
delete mode 100644 Basics/adjoint.ipynb
diff --git a/Basics/adjoint.ipynb b/Basics/adjoint.ipynb
deleted file mode 100644
index 1b7bd03..0000000
--- a/Basics/adjoint.ipynb
+++ /dev/null
@@ -1,84 +0,0 @@
-{
- "cells": [
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "**1.**\n",
- "### $A.adj(A)= adj(A).A = |A|I$\n",
- "\n",
- "#### Let A be a square matrix of order n and C represent the cofactor matrix of A and adj(A) represent the adjoint of matrix A .\n",
- "\n",
- "$$(adj(A))_{ij} = [C_{ij}]^T = C_{ji}$$\n",
- "
\n",
- "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.(adj(A))_{rj}$$\n",
- "
\n",
- "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr}$$\n",
- "
\n",
- "#### We Know,\n",
- "\n",
- "##### Sum of products of elements with their corresponding cofactor is equal to the determinant of the matrix.\n",
- "
\n",
- "\n",
- "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr} = \\left\\{\\begin{array}\\\\ |A| & \\mbox{if} \\ i=j \n",
- " \\\\ 0 & \\mbox{if} \\ i \\neq j\n",
- "\\end{array}\\right$$\n",
- "\n",
- "
\n",
- "\n",
- "#### So,\n",
- "
\n",
- "$$ A.adj(A) = \\begin{bmatrix} |A| & 0 & 0 & . & . & 0\n",
- " \\\\ 0 & |A| & 0 & . & . & 0\n",
- " \\\\ . & . & . & . & . & .\n",
- " \\\\ . & . & . & . & . & .\n",
- " \\\\ 0 & 0 & . & . & . & |A|\n",
- " \\end{bmatrix}_{n\\times n}$$\n",
- " \n",
- "
\n",
- "$$ A.adj(A) = |A|.\\begin{bmatrix} 1 & 0 & 0 & . & . & 0\n",
- " \\\\ 0 & 1 & 0 & . & . & 0\n",
- " \\\\ . & . & . & . & . & .\n",
- " \\\\ . & . & . & . & . & .\n",
- " \\\\ 0 & 0 & . & . & . & 1\n",
- " \\end{bmatrix}_{n\\times n}$$\n",
- " \n",
- "
\n",
- "$$A.adj(A) = |A|.I_{n}$$\n",
- "\n",
- "\n",
- "##### Similarly we can prove adj(A).A = |A|.I\n",
- "\n",
- "\n"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": []
- }
- ],
- "metadata": {
- "kernelspec": {
- "display_name": "Python 3",
- "language": "python",
- "name": "python3"
- },
- "language_info": {
- "codemirror_mode": {
- "name": "ipython",
- "version": 3
- },
- "file_extension": ".py",
- "mimetype": "text/x-python",
- "name": "python",
- "nbconvert_exporter": "python",
- "pygments_lexer": "ipython3",
- "version": "3.7.3"
- }
- },
- "nbformat": 4,
- "nbformat_minor": 2
-}
From 818ec9d8de57fd0f5d4276d86262e73ddcf681c6 Mon Sep 17 00:00:00 2001
From: rockysaikia730 <67681828+rockysaikia730@users.noreply.github.com>
Date: Wed, 7 Oct 2020 22:21:53 +0530
Subject: [PATCH 13/18] Adjoint
---
Basics/adjoint.ipynb | 84 ++++++++++++++++++++++++++++++++++++++++++++
1 file changed, 84 insertions(+)
create mode 100644 Basics/adjoint.ipynb
diff --git a/Basics/adjoint.ipynb b/Basics/adjoint.ipynb
new file mode 100644
index 0000000..682069f
--- /dev/null
+++ b/Basics/adjoint.ipynb
@@ -0,0 +1,84 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "**1.**\n",
+ "### $A.adj(A)= adj(A).A = |A|I$\n",
+ "\n",
+ "#### Let A be a square matrix of order n and C represent the cofactor matrix of A and adj(A) represent the adjoint of matrix A .\n",
+ "\n",
+ "$$(adj(A))_{ij} = [C_{ij}]^T = C_{ji}$$\n",
+ "
\n",
+ "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.(adj(A))_{rj}$$\n",
+ "
\n",
+ "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr}$$\n",
+ "
\n",
+ "#### We Know,\n",
+ "\n",
+ "##### Sum of products of elements with their corresponding cofactor is equal to the determinant of the matrix.\n",
+ "
\n",
+ "\n",
+ "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr} = \\left\\{\\begin{array}\\\\ |A| & \\mbox{if} \\ i=j \n",
+ " \\\\ 0 & \\mbox{if} \\ i \\neq j\n",
+ " \\end{array}\\right.$$\n",
+ "\n",
+ "
\n",
+ "\n",
+ "#### So,\n",
+ "
\n",
+ "$$ A.adj(A) = \\begin{bmatrix} |A| & 0 & 0 & . & . & 0\n",
+ " \\\\ 0 & |A| & 0 & . & . & 0\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ 0 & 0 & . & . & . & |A|\n",
+ " \\end{bmatrix}_{n\\times n}$$\n",
+ " \n",
+ "
\n",
+ "$$ A.adj(A) = |A|.\\begin{bmatrix} 1 & 0 & 0 & . & . & 0\n",
+ " \\\\ 0 & 1 & 0 & . & . & 0\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ 0 & 0 & . & . & . & 1\n",
+ " \\end{bmatrix}_{n\\times n}$$\n",
+ " \n",
+ "
\n",
+ "$$A.adj(A) = |A|.I_{n}$$\n",
+ "\n",
+ "\n",
+ "##### Similarly we can prove adj(A).A = |A|.I\n",
+ "\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.7.3"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
From 975c7260de462db230733f488aec721093266431 Mon Sep 17 00:00:00 2001
From: rockysaikia730 <67681828+rockysaikia730@users.noreply.github.com>
Date: Wed, 7 Oct 2020 22:25:13 +0530
Subject: [PATCH 14/18] Update adjoint.ipynb
---
Basics/adjoint.ipynb | 4 ++--
1 file changed, 2 insertions(+), 2 deletions(-)
diff --git a/Basics/adjoint.ipynb b/Basics/adjoint.ipynb
index 682069f..ff0be9b 100644
--- a/Basics/adjoint.ipynb
+++ b/Basics/adjoint.ipynb
@@ -20,9 +20,9 @@
"##### Sum of products of elements with their corresponding cofactor is equal to the determinant of the matrix.\n",
"
\n",
"\n",
- "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr} = \\left\\{\\begin{array}\\\\ |A| & \\mbox{if} \\ i=j \n",
+ "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr} = \\left\\(\\begin{array}\\\\ |A| & \\mbox{if} \\ i=j \n",
" \\\\ 0 & \\mbox{if} \\ i \\neq j\n",
- " \\end{array}\\right.$$\n",
+ " \\end{array}\\right.)$$\n",
"\n",
"
\n",
"\n",
From 78792f714b37cf56b9b767c67e9befca6d9e41f9 Mon Sep 17 00:00:00 2001
From: rockysaikia730 <67681828+rockysaikia730@users.noreply.github.com>
Date: Wed, 7 Oct 2020 22:28:00 +0530
Subject: [PATCH 15/18] Update adjoint.ipynb
---
Basics/adjoint.ipynb | 4 ++--
1 file changed, 2 insertions(+), 2 deletions(-)
diff --git a/Basics/adjoint.ipynb b/Basics/adjoint.ipynb
index ff0be9b..676b9bb 100644
--- a/Basics/adjoint.ipynb
+++ b/Basics/adjoint.ipynb
@@ -20,9 +20,9 @@
"##### Sum of products of elements with their corresponding cofactor is equal to the determinant of the matrix.\n",
"
\n",
"\n",
- "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr} = \\left\\(\\begin{array}\\\\ |A| & \\mbox{if} \\ i=j \n",
+ "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr} = \\left\\{\\begin{array}\\\\ |A| & \\mbox{if} \\ i=j \n",
" \\\\ 0 & \\mbox{if} \\ i \\neq j\n",
- " \\end{array}\\right.)$$\n",
+ " \\end{array}\\right.}$$\n",
"\n",
"
\n",
"\n",
From b5d2e1f4c0db934ec8691973749b34f6a7f6d0f2 Mon Sep 17 00:00:00 2001
From: rockysaikia730 <67681828+rockysaikia730@users.noreply.github.com>
Date: Wed, 7 Oct 2020 22:31:20 +0530
Subject: [PATCH 16/18] Update adjoint.ipynb
---
Basics/adjoint.ipynb | 3 +--
1 file changed, 1 insertion(+), 2 deletions(-)
diff --git a/Basics/adjoint.ipynb b/Basics/adjoint.ipynb
index 676b9bb..9cf2ce9 100644
--- a/Basics/adjoint.ipynb
+++ b/Basics/adjoint.ipynb
@@ -22,8 +22,7 @@
"\n",
"$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr} = \\left\\{\\begin{array}\\\\ |A| & \\mbox{if} \\ i=j \n",
" \\\\ 0 & \\mbox{if} \\ i \\neq j\n",
- " \\end{array}\\right.}$$\n",
- "\n",
+ " \\end{array}\\right.$$\n",
"
\n",
"\n",
"#### So,\n",
From 431c544c41b3e2ffdfac3d48e3de4c012b2a5243 Mon Sep 17 00:00:00 2001
From: rockysaikia730 <67681828+rockysaikia730@users.noreply.github.com>
Date: Wed, 7 Oct 2020 22:38:46 +0530
Subject: [PATCH 17/18] Delete adjoint.ipynb
---
Basics/adjoint.ipynb | 83 --------------------------------------------
1 file changed, 83 deletions(-)
delete mode 100644 Basics/adjoint.ipynb
diff --git a/Basics/adjoint.ipynb b/Basics/adjoint.ipynb
deleted file mode 100644
index 9cf2ce9..0000000
--- a/Basics/adjoint.ipynb
+++ /dev/null
@@ -1,83 +0,0 @@
-{
- "cells": [
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "**1.**\n",
- "### $A.adj(A)= adj(A).A = |A|I$\n",
- "\n",
- "#### Let A be a square matrix of order n and C represent the cofactor matrix of A and adj(A) represent the adjoint of matrix A .\n",
- "\n",
- "$$(adj(A))_{ij} = [C_{ij}]^T = C_{ji}$$\n",
- "
\n",
- "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.(adj(A))_{rj}$$\n",
- "
\n",
- "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr}$$\n",
- "
\n",
- "#### We Know,\n",
- "\n",
- "##### Sum of products of elements with their corresponding cofactor is equal to the determinant of the matrix.\n",
- "
\n",
- "\n",
- "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr} = \\left\\{\\begin{array}\\\\ |A| & \\mbox{if} \\ i=j \n",
- " \\\\ 0 & \\mbox{if} \\ i \\neq j\n",
- " \\end{array}\\right.$$\n",
- "
\n",
- "\n",
- "#### So,\n",
- "
\n",
- "$$ A.adj(A) = \\begin{bmatrix} |A| & 0 & 0 & . & . & 0\n",
- " \\\\ 0 & |A| & 0 & . & . & 0\n",
- " \\\\ . & . & . & . & . & .\n",
- " \\\\ . & . & . & . & . & .\n",
- " \\\\ 0 & 0 & . & . & . & |A|\n",
- " \\end{bmatrix}_{n\\times n}$$\n",
- " \n",
- "
\n",
- "$$ A.adj(A) = |A|.\\begin{bmatrix} 1 & 0 & 0 & . & . & 0\n",
- " \\\\ 0 & 1 & 0 & . & . & 0\n",
- " \\\\ . & . & . & . & . & .\n",
- " \\\\ . & . & . & . & . & .\n",
- " \\\\ 0 & 0 & . & . & . & 1\n",
- " \\end{bmatrix}_{n\\times n}$$\n",
- " \n",
- "
\n",
- "$$A.adj(A) = |A|.I_{n}$$\n",
- "\n",
- "\n",
- "##### Similarly we can prove adj(A).A = |A|.I\n",
- "\n",
- "\n"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": null,
- "metadata": {},
- "outputs": [],
- "source": []
- }
- ],
- "metadata": {
- "kernelspec": {
- "display_name": "Python 3",
- "language": "python",
- "name": "python3"
- },
- "language_info": {
- "codemirror_mode": {
- "name": "ipython",
- "version": 3
- },
- "file_extension": ".py",
- "mimetype": "text/x-python",
- "name": "python",
- "nbconvert_exporter": "python",
- "pygments_lexer": "ipython3",
- "version": "3.7.3"
- }
- },
- "nbformat": 4,
- "nbformat_minor": 2
-}
From 168db6f4fe0015c89347bb7ff9adba3e084c7adf Mon Sep 17 00:00:00 2001
From: rockysaikia730 <67681828+rockysaikia730@users.noreply.github.com>
Date: Wed, 7 Oct 2020 22:39:25 +0530
Subject: [PATCH 18/18] Adjoint
---
Basics/adjoint.ipynb | 86 ++++++++++++++++++++++++++++++++++++++++++++
1 file changed, 86 insertions(+)
create mode 100644 Basics/adjoint.ipynb
diff --git a/Basics/adjoint.ipynb b/Basics/adjoint.ipynb
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+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "**1.**\n",
+ "### $A.adj(A)= adj(A).A = |A|I$\n",
+ "\n",
+ "#### Let A be a square matrix of order n and C represent the cofactor matrix of A and adj(A) represent the adjoint of matrix A .\n",
+ "\n",
+ "$$(adj(A))_{ij} = [C_{ij}]^T = C_{ji}$$\n",
+ "
\n",
+ "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.(adj(A))_{rj}$$\n",
+ "
\n",
+ "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr}$$\n",
+ "
\n",
+ "#### We Know,\n",
+ "\n",
+ "##### Sum of products of elements with their corresponding cofactor is equal to the determinant of the matrix.\n",
+ "
\n",
+ "\n",
+ "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr} = \\left\\{\\begin{array} \\\\ \n",
+ " |A| & \\mbox{if} \\ i=j \\\\ \n",
+ " 0 & \\mbox{if} \\ i \\neq j\n",
+ " \\end{array}\n",
+ "\\right.$$\n",
+ "\n",
+ "
\n",
+ "\n",
+ "#### So,\n",
+ "
\n",
+ "$$ A.adj(A) = \\begin{bmatrix} |A| & 0 & 0 & . & . & 0\n",
+ " \\\\ 0 & |A| & 0 & . & . & 0\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ 0 & 0 & . & . & . & |A|\n",
+ " \\end{bmatrix}_{n\\times n}$$\n",
+ " \n",
+ "
\n",
+ "$$ A.adj(A) = |A|.\\begin{bmatrix} 1 & 0 & 0 & . & . & 0\n",
+ " \\\\ 0 & 1 & 0 & . & . & 0\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ . & . & . & . & . & .\n",
+ " \\\\ 0 & 0 & . & . & . & 1\n",
+ " \\end{bmatrix}_{n\\times n}$$\n",
+ " \n",
+ "
\n",
+ "$$A.adj(A) = |A|.I_{n}$$\n",
+ "\n",
+ "\n",
+ "##### Similarly we can prove adj(A).A = |A|.I\n",
+ "\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
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+ "file_extension": ".py",
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