Find the inverse of A = [[2, −1], [−3, 4]]

Question

ScanSolve AI Tutor · Accepted Answer

Here's how to find the inverse of matrix $A$ :

Given matrix $A = \begin{bmatrix} 2 & -1 \\ -3 & 4 \end{bmatrix}$ .

Calculate the determinant of matrix $A$ .
For a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ , the determinant is $ad-bc$ .
$\det(A) = (2)(4) - (-1)(-3)$ $\det(A) = 8 - 3$ $\det(A) = 5$
Find the adjugate matrix (also known as the adjoint matrix).
For a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ , the adjugate is $\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ .
$\text{adj}(A) = \begin{bmatrix} 4 & -(-1) \\ -(-3) & 2 \end{bmatrix}$ $\text{adj}(A) = \begin{bmatrix} 4 & 1 \\ 3 & 2 \end{bmatrix}$
Calculate the inverse matrix $A^{-1}$ .
The formula for the inverse of a $2 \times 2$ matrix is $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$ .
$A^{-1} = \frac{1}{5} \begin{bmatrix} 4 & 1 \\ 3 & 2 \end{bmatrix}$
Multiply the scalar $\frac{1}{5}$ by each element in the adjugate matrix.
$A^{-1} = \begin{bmatrix} \frac{4}{5} & \frac{1}{5} \\ \frac{3}{5} & \frac{2}{5} \end{bmatrix}$
The final answer is $\boxed{\begin{bmatrix} \frac{4}{5} & \frac{1}{5} \\ \frac{3}{5} & \frac{2}{5} \end{bmatrix}}$ .

Find the inverse of A = [[2, -1], [-3, 4]]