Question

(b) If \( A = \begin{bmatrix} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{bmatrix} \), find \( A^{-1} \).

Hint:

The inverse of a matrix exists only if its determinant is non-zero. Use cofactor expansion for determinant and adjugate calculations.

Answer 1

To find \( A^{-1} \), compute: \[ A^{-1} = \frac{1}{\det(A)} \text{adj}(A), \] where \( \det(A) \) is the determinant of \( A \) and \( \text{adj}(A) \) is the adjugate matrix of \( A \).

Step 1: Compute \( \det(A) \)

\[ \det(A) = 2\begin{vmatrix} 2 & -4 \\ 1 & -2 \end{vmatrix} - (-3)\begin{vmatrix} 3 & -4 \\ 1 & -2 \end{vmatrix} + 5\begin{vmatrix} 3 & 2 \\ 1 & 1 \end{vmatrix}. \] Simplify to find \( \det(A) \).

Step 2: Compute \( \text{adj}(A) \)

Find the cofactor matrix of \( A \) and transpose it to get \( \text{adj}(A) \).

Step 3: Compute \( A^{-1} \)

\[ A^{-1} = \frac{1}{\det(A)} \text{adj}(A). \]