आव्यूह के सहखंडज और व्युत्क्रम: Difference between revisions

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=== <math>3 \ X \ 3</math> आव्यूह का सहखंडज ===
=== <math>3 \ X \ 3</math> आव्यूह का सहखंडज ===
<math>A = \begin{bmatrix} 2 & -1 & 3 \\ 0 & 5 & 2 \\ 1 & -1 & -2 \end{bmatrix}</math>
'''Step 1:''' Find the minor matrix <math>M</math> of all the elements of matrix <math>A</math>.
'''Step 1:''' Find the minor matrix <math>M</math> of all the elements of matrix <math>A</math>.
'''''Row 1:'''''
Minor of <math>2 = \begin{vmatrix} 5 & 2 \\ -1 & -2 \end{vmatrix}=(-10-(-2))=-10+2=-8</math>
Minor of <math>-1 = \begin{vmatrix} 0 & 2 \\ 1 & -2 \end{vmatrix}=(0-2)=-2</math>
Minor of <math>3 = \begin{vmatrix} 0 & 5 \\ 1 & -1 \end{vmatrix}=(0-5))=-5</math>
'''''Row 2:'''''
Minor of <math>0 = \begin{vmatrix} -1 & 3 \\ -1 & -2 \end{vmatrix}=(2-(-3))=2+3=5</math>
Minor of <math>5 = \begin{vmatrix} 2 & 3 \\ 1 & -2 \end{vmatrix}=(-4-3)=-7</math>
Minor of <math>2 = \begin{vmatrix} 2 & -1 \\ 1 & -1 \end{vmatrix}=(-2-(-1))=-1</math>
'''''Row 3:'''''
Minor of <math>1 = \begin{vmatrix} -1 & 3 \\ 5 & 2 \end{vmatrix}=(-2-15)=-17</math>
Minor of <math>-1 = \begin{vmatrix} 2 & 3 \\ 0 & 2 \end{vmatrix}=(4-0)=4</math>
Minor of <math>-2 = \begin{vmatrix} 2 & -1 \\ 0 & 5 \end{vmatrix}=(10-0)=10</math>
Minor of Matrix <math>A</math> is <math>M= \begin{bmatrix} -8 & -2 & -5 \\ 5 & -7 & -1 \\ -17 & 4 & 10 \end{bmatrix}</math>
'''Step 2:''' Find the cofactor matrix <math>C</math> of all the minor elements of matrix <math>M</math>
To find the cofactors of <math>3 \ X \ 3</math> matrix, the corresponding minors should be multiplied by the signs below according to their position.
<math>C= \begin{bmatrix} + & - & + \\ - & + & - \\ + & - & + \end{bmatrix}</math>
Minor of Matrix <math>A</math> is <math>M= \begin{bmatrix} -8 & -2 & -5 \\ 5 & -7 & -1 \\ -17 & 4 & 10 \end{bmatrix}</math>
Cofactor of Matrix A is <math>C= \begin{bmatrix} -8 & 2 & -5 \\ -5 & -7 & 1 \\ -17 & -4 & 10 \end{bmatrix}</math>
'''Step 3:''' Find the adj <math>A</math> by taking the transpose of the cofactor matrix <math>C</math>
Adjoint of Matrix A is adj <math>A</math> = Transpose of the Cofactor Matrix <math>C</math> <math>= \begin{bmatrix} -8 & -5 & -17 \\ 2 & -7 & -4 \\ -5 & 1 & 10 \end{bmatrix}</math>
== Inverse of a Matrix ==
The inverse of a matrix <math>A</math>, which is represented as <math>A^{-1}</math>, is found using the adjoint of a matrix.
A<sup>-1</sup> = (1/|A|) × adj(A). Here, <math>A^{-1}=\left [\frac{1}{\begin{vmatrix} A \end{vmatrix}}\right ] \times adj(A)</math>
Here
* <math>\begin{vmatrix} A \end{vmatrix}</math>= the determinant of <math>A</math>
* <math>adj(A)</math>= adjoint of <math>A</math>
=== Inverse of a <math>3 \ X \ 3</math> Matrix ===
determinant of <math>A =</math>  <math>{\begin{vmatrix} A \end{vmatrix}} = \begin{vmatrix} 2 & -1 & 3 \\ 0 & 5 & 2 \\ 1 & -1 & -2 \end{vmatrix}=2(-10-(-2))-(-1)(0-2)+3(0-5)=2(-8)-2-15=-33</math>
Adjoint of Matrix <math>A =</math> <math>adj(A)</math> <math>= \begin{bmatrix} -8 & -5 & -17 \\ 2 & -7 & -4 \\ -5 & 1 & 10 \end{bmatrix}</math>
Inverse of matrix <math>A =</math><math>A^{-1}=\left [\frac{1}{\begin{vmatrix} A \end{vmatrix}}\right ] \times adj(A)</math>
<math>A^{-1}=\left [\frac{1}{-33}\right ] \times  \begin{bmatrix} -8 & -5 & -17 \\ 2 & -7 & -4 \\ -5 & 1 & 10 \end{bmatrix}=\begin{bmatrix} \frac{8}{33} & \frac{5}{33} & \frac{17}{33} \\ - \frac{2}{33} & \frac{7}{33} & \frac{4}{33} \\ \frac{5}{33} & - \frac{1}{33} &
- \frac{10}{33} \end{bmatrix}</math>
[[Category:सारणिक]][[Category:गणित]][[Category:कक्षा-12]]
[[Category:सारणिक]][[Category:गणित]][[Category:कक्षा-12]]

Revision as of 11:41, 8 February 2024

किसी आव्यूह के व्युत्क्रम की गणना करने के लिए आव्यूह के सहखंडज की आवश्यकता होती है।

आव्यूह के सहखंडज

आव्यूह का सहखंडज, के सहखंड आव्यूह का परिवर्त है। वर्ग आव्यूह का सहखंडज (adj.) द्वारा निरूपित किया जाता है। मान लीजिए , कोटि का एक वर्ग आव्यूह है।

किसी आव्यूह का सहखंडज ज्ञात करने में सम्मिलित प्रक्रिया इस प्रकार हैं:

  • आव्यूह के सभी अवयवों का उपसारणिक आव्यूह को ज्ञात करें ।
  • आव्यूह के सभी उपसारणिक अवयवों का सहखंड आव्यूह को ज्ञात करें ।
  • सहखंड आव्यूह का परिवर्त लेते हुए सहखंडज (adj.) को ज्ञात करें ।

आव्यूह का सहखंडज

Step 1: Find the minor matrix of all the elements of matrix .

Step 1: Find the minor matrix of all the elements of matrix .

Row 1:

Minor of

Minor of

Minor of

Row 2:

Minor of

Minor of

Minor of

Row 3:

Minor of

Minor of

Minor of

Minor of Matrix is


Step 2: Find the cofactor matrix of all the minor elements of matrix

To find the cofactors of matrix, the corresponding minors should be multiplied by the signs below according to their position.

Minor of Matrix is

Cofactor of Matrix A is


Step 3: Find the adj by taking the transpose of the cofactor matrix

Adjoint of Matrix A is adj = Transpose of the Cofactor Matrix

Inverse of a Matrix

The inverse of a matrix , which is represented as , is found using the adjoint of a matrix.

A-1 = (1/|A|) × adj(A). Here,

Here

  • = the determinant of
  • = adjoint of

Inverse of a Matrix

determinant of

Adjoint of Matrix

Inverse of matrix