Adjoint and Inverse of a Matrix — Maths STD 12 Science — Question

Let $\text{A}=\begin{bmatrix}5 & 4 \\1 & 1 \end{bmatrix}\text{ and B}=\begin{bmatrix}1 & -2 \\1 & 3 \end{bmatrix}$

So, AX = B

or X = A^-1B .....(i)

$|\text{A}|=1\neq0$

Cofactors of A are:

C₁₁ = 1, C₁₂ = -1

C₂₁ = -4, C₂₂ = 5

$\therefore\ \text{adj A}=\begin{bmatrix}\text{C}_{11} & \text{C}_{12} \\ \text{C}_{21} & \text{C}_{22} \end{bmatrix}$

$=\begin{bmatrix}1 & -1 \\-4 & 5 \end{bmatrix}^\text{T}$

$=\begin{bmatrix}1 & -4 \\-1 & 5 \end{bmatrix}$

Now, $\text{A}^{-1}=\frac{1}{|\text{A}|}\text{ adj A}$

$\text{A}^{1}=\frac{1}{2}=\begin{bmatrix}1 & -4 \\-1 & 5 \end{bmatrix}$

So from (i)

$\text{X}=\begin{bmatrix}1 & -4 \\-1 & 5 \end{bmatrix}=\begin{bmatrix}1 & 2 \\ 1 & 3 \end{bmatrix}$

$\text{X}=\begin{bmatrix}-3 & -14 \\ 4 & 17 \end{bmatrix}$