4 5 Adjoint And Inverse – Class 11TH Math

4.5 Adjoint and Inverse of a Square Matrix

For $A = [a_{ij}]_{3 \times 3}$ with cofactors $A_{ij}$ , the adjoint is the transpose of the cofactor matrix:

\operatorname{adj} A = \begin{bmatrix} A_{11} & A_{21} & A_{31} \\ A_{12} & A_{22} & A_{32} \\ A_{13} & A_{23} & A_{33} \end{bmatrix}

If $A$ is nonsingular ( $|A| \neq 0$ ) and $AB = BA = I$ , then $B = A^{-1}$ . Equivalently

A^{-1} = \dfrac{1}{|A|}\operatorname{adj} A

Find $A^{-1}$ if $A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 1 & -1 & 1 \end{bmatrix}$ .

Solution

Cofactors:

\begin{align*} A_{11} &= 3,\ A_{12} = 1,\ A_{13} = -2, \\ A_{21} &= -2,\ A_{22} = -1,\ A_{23} = 1, \\ A_{31} &= -4,\ A_{32} = -1,\ A_{33} = 2 \end{align*}

\operatorname{adj} A = \begin{bmatrix} 3 & -2 & -4 \\ 1 & -1 & -1 \\ -2 & 1 & 2 \end{bmatrix}, \quad |A| = 1(3) + 0 + 2(-2) = -1

A^{-1} = \dfrac{1}{-1}\operatorname{adj} A = \begin{bmatrix} -3 & 2 & 4 \\ -1 & 1 & 1 \\ 2 & -1 & -2 \end{bmatrix}