Question

If \( A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} \) and \( A^2 + 7I = kA \), then the value of \( k \) is:

• A. 1
• B. 2
• C. 5
• D. 7

Hint:

When finding \( A^2 \) and working with matrix equations, always ensure the matrix dimensions align for addition and scalar multiplication.

Answer 1

The Correct Option is C

Step 1: Compute \( A^2 \). This is done by multiplying matrix \( A \) by itself: \[ A^2 = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} = \begin{bmatrix} 8 & 5 \\ 1 & 3 \end{bmatrix} \]
Step 2: Substitute \( A^2 \) into the equation \( A^2 + 7I = kA \). The equation becomes: \[ \begin{bmatrix} 8 & 5 \\ 1 & 3 \end{bmatrix} + 7 \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = k \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} \] Simplify the left side: \[ \begin{bmatrix} 8 & 5 \\ 1 & 3 \end{bmatrix} + \begin{bmatrix} 7 & 0 \\ 0 & 7 \end{bmatrix} = \begin{bmatrix} 15 & 5 \\ 1 & 10 \end{bmatrix} \] Equate this to \( kA \): \[ \begin{bmatrix} 15 & 5 \\ 1 & 10 \end{bmatrix} = k \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} \] By equating corresponding elements, we find \( k = 5 \). 
Step 3: Verify the options. The calculated value \( k = 5 \) matches option (C).