Question

If \( A = \begin{bmatrix} 3 & 7 \\ 2 & 5 \end{bmatrix} \), then \( A^2 (\text{adj} A) \) is:

• A. \( I \)
• B. \( 4I \)
• C. \( 2A \)
• D. \( 3A \)
• E. \( A \)

Hint:

When \( A^2 ({adj} A) = \det(A) \times A \), use the determinant of \( A \) to simplify the calculation.

Answer 1

Answer: (E)

The adjugate of \( A \), denoted \( \text{adj}(A) \), is given by the formula: \[ \text{adj}(A) = \begin{bmatrix} 5 & -7 \\ -2 & 3 \end{bmatrix} \] Now, \( A^2 (\text{adj} A) = A \), as per the property of matrices where \( A^2 \times \text{adj}(A) = \det(A) \times A \). 
Here, \( \det(A) = (3 \times 5) - (7 \times 2) = 15 - 14 = 1 \), so the result is \( A \).