find the adjoint of matrices

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ScanSolve AI Tutor · Accepted Answer

To find the adjoint of the given matrix, we first need to find the cofactor matrix and then transpose it.

The given matrix is:

A = \begin{bmatrix} 0 & 1 & 2 \\ 0 & 1 & 1 \\ 1 & 0 & 2 \end{bmatrix}

The cofactor $C_{ij}$ of an element $a_{ij}$ is given by $C_{ij} = (-1)^{i+j} M_{ij}$ , where $M_{ij}$ is the minor of $a_{ij}$ . The adjoint of matrix $A$ is the transpose of its cofactor matrix, i.e., $\text{adj}(A) = (C_{ij})^T$ .

Calculate the Minors ( $M_{ij}$ ) for each element.
- $M_{11} = \begin{vmatrix} 1 & 1 \\ 0 & 2 \end{vmatrix} = (1)(2) - (1)(0) = 2 - 0 = 2$
- $M_{12} = \begin{vmatrix} 0 & 1 \\ 1 & 2 \end{vmatrix} = (0)(2) - (1)(1) = 0 - 1 = -1$
- $M_{13} = \begin{vmatrix} 0 & 1 \\ 1 & 0 \end{vmatrix} = (0)(0) - (1)(1) = 0 - 1 = -1$
- $M_{21} = \begin{vmatrix} 1 & 2 \\ 0 & 2 \end{vmatrix} = (1)(2) - (2)(0) = 2 - 0 = 2$
- $M_{22} = \begin{vmatrix} 0 & 2 \\ 1 & 2 \end{vmatrix} = (0)(2) - (2)(1) = 0 - 2 = -2$
- $M_{23} = \begin{vmatrix} 0 & 1 \\ 1 & 0 \end{vmatrix} = (0)(0) - (1)(1) = 0 - 1 = -1$
- $M_{31} = \begin{vmatrix} 1 & 2 \\ 1 & 1 \end{vmatrix} = (1)(1) - (2)(1) = 1 - 2 = -1$
- $M_{32} = \begin{vmatrix} 0 & 2 \\ 0 & 1 \end{vmatrix} = (0)(1) - (2)(0) = 0 - 0 = 0$
- $M_{33} = \begin{vmatrix} 0 & 1 \\ 0 & 1 \end{vmatrix} = (0)(1) - (1)(0) = 0 - 0 = 0$
Calculate the Cofactors ( $C_{ij}$ ) for each element.
- $C_{11} = (-1)^{1+1} M_{11} = (1)(2) = 2$
- $C_{12} = (-1)^{1+2} M_{12} = (-1)(-1) = 1$
- $C_{13} = (-1)^{1+3} M_{13} = (1)(-1) = -1$
- $C_{21} = (-1)^{2+1} M_{21} = (-1)(2) = -2$
- $C_{22} = (-1)^{2+2} M_{22} = (1)(-2) = -2$
- $C_{23} = (-1)^{2+3} M_{23} = (-1)(-1) = 1$
- $C_{31} = (-1)^{3+1} M_{31} = (1)(-1) = -1$
- $C_{32} = (-1)^{3+2} M_{32} = (-1)(0) = 0$
- $C_{33} = (-1)^{3+3} M_{33} = (1)(0) = 0$
Form the Cofactor Matrix $C$ .
$C = \begin{bmatrix} 2 & 1 & -1 \\ -2 & -2 & 1 \\ -1 & 0 & 0 \end{bmatrix}$
Find the Adjoint of the matrix $A$ by taking the transpose of the Cofactor Matrix $C$ .
$\text{adj}(A) = C^T = \begin{bmatrix} 2 & -2 & -1 \\ 1 & -2 & 0 \\ -1 & 1 & 0 \end{bmatrix}$
The adjoint of the given matrix is:
$\boxed{\begin{bmatrix} 2 & -2 & -1 \\ 1 & -2 & 0 \\ -1 & 1 & 0 \end{bmatrix}}$

Find the adjoint of matrices

Step-by-step solution

Calculate the Minors ( $M_{ij}$ ) for each element.

Calculate the Cofactors ( $C_{ij}$ ) for each element.

Form the Cofactor Matrix $C$ .

Find the Adjoint of the matrix $A$ by taking the transpose of the Cofactor Matrix $C$ .

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More Mathematics Questions

Find the adjoint of matrices

Step-by-step solution

Calculate the Minors (MijM_{ij}Mij​) for each element.

Calculate the Cofactors (CijC_{ij}Cij​) for each element.

Form the Cofactor Matrix CCC.

Find the Adjoint of the matrix AAA by taking the transpose of the Cofactor Matrix CCC.