Difference between revisions of "Adjoint and Inverse of a Matrix"
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Ramamurthy S (talk | contribs) |
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=== Inverse of a <math>3 \ X \ 3</math> Matrix === | === Inverse of a <math>3 \ X \ 3</math> Matrix === | ||
determinant of <math>A = \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> | 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> | |||
<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> |
Revision as of 08:35, 6 February 2024
The adjoint of a matrix is needed to calculate the inverse of a matrix.
Adjoint of a Matrix
The adjoint of a matrix is the transpose of the cofactor matrix of . The adjoint of a square matrix is denoted by adj . Let be a square matrix of order .
Steps involved in finding the adjoint of a matrix are:
- Find the minor matrix of all the elements of matrix .
- Find the cofactor matrix of all the minor elements of matrix
- Find the adj by taking the transpose of the cofactor matrix .
Adjoint of a 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
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