Question

If \[ A = \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix} \quad \text{and} \quad A^2 = kA, \quad \text{then find the value of k.} \]

Hint:

When solving matrix equations like $ A^2 = kA $, perform matrix multiplication first and then equate the corresponding elements to find the unknowns.

Answer 1

We are given that $ A = \begin{bmatrix} 2 & -2
-2 & 2 \end{bmatrix} $ and $ A^2 = kA $. Step 1: Calculate $ A^2 $ by multiplying matrix $ A $ by itself: \[ A^2 = A \times A = \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix} \times \begin{bmatrix} 2 & -2 \ -2 \& 2 \end{bmatrix} \] Perform the matrix multiplication: \[ A^2 = \begin{bmatrix} (2 \times 2 + -2 \times -2) & (2 \times -2 + -2 \times 2) \\ (-2 \times 2 + 2 \times -2) & (-2 \times -2 + 2 \times 2) \end{bmatrix} \] \[ A^2 = \begin{bmatrix} 4 + 4 & -4 - 4 \\ -4 - 4 & 4 + 4 \end{bmatrix} \] \[ A^2 = \begin{bmatrix} 8 & -8
-8 & 8 \end{bmatrix} \] Step 2: Compare $ A^2 $ with $ kA $: We are given that $ A^2 = kA $, so: \[ \begin{bmatrix} 8 & -8 \ -8 &\ 8 \end{bmatrix} = k \begin{bmatrix} 2 & -2
-2 & 2 \end{bmatrix} \] Step 3: Set up equations for each element: \[ 8 = 2k \quad \text{and} \quad -8 = -2k \] Step 4: Solve for $ k $: From the first equation: \[ k = \frac{8}{2} = 4 \] Thus, the value of $ k $ is $ \boxed{4} $.