Question

Find the inverse of matrix \( A \).

Hint:

To find the inverse of a 2x2 matrix: 1. Compute the determinant \( {det}(A) = ad - bc \). 2. Swap \( a \) and \( d \), and change the signs of \( b \) and \( c \) in the adjoint. 3. Divide each element by \( {det}(A) \).

Answer 1

Matrix \( A \): \[ A = \begin{bmatrix} 10 & -8 \\ -10 & 16 \end{bmatrix}. \]
Step 1: Calculate the determinant of \( A \): \[ \det(A) = (10)(16) - (-8)(-10) = 160 - 80 = 80. \]
Step 2: Calculate the adjoint of \( A \): \[ \text{Adj}(A) = \begin{bmatrix} 16 & 8 \\ 10 & 10 \end{bmatrix}. \]
Step 3: Calculate the inverse of \( A \): \[ A^{-1} = \frac{1}{\det(A)} \cdot \text{Adj}(A) = \frac{1}{80} \cdot \begin{bmatrix} 16 & 8 \\ 10 & 10 \end{bmatrix}. \]
Simplify: \[ A^{-1} = \begin{bmatrix} 0.2 & 0.1 \\ 0.125 & 0.125 \end{bmatrix}. \]