Question

If \( A = \left[ \begin{array}{cc} 1 & -1 \\ 7 & 0 \end{array} \right] \), \( |A| = \left| \begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array} \right| \), and \( A^2 = 8A + kI \), then the value of \( k \) is

• A. \( \frac{1}{7} \)
• B. \( -\frac{1}{7} \)
• C. -7
• D. 7

Hint:

When working with matrix equations, multiply the matrices as required and solve for unknown constants by equating the resulting matrix elements.

Answer 1

The Correct Option is D

Step 1: Use the given matrix and condition.
We are given \( A^2 = 8A + kI \). From the equation \( A^2 = 8A + kI \), we can equate the matrix elements to find \( k \).
Step 2: Compute \( A^2 \).
Find \( A^2 \) by multiplying the matrix \( A \) by itself: \[ A^2 = \left[ \begin{array}{cc} 1 & -1 \\7 & 0 \end{array} \right] \times \left[ \begin{array}{cc} 1 & -1 \\ 7 & 0 \end{array} \right] \] Perform the matrix multiplication to get: \[ A^2 = \left[ \begin{array}{cc} 8 & -8 \\ 7 & -7 \end{array} \right] \]
Step 3: Use the condition \( A^2 = 8A + kI \).
Using the given condition \( A^2 = 8A + kI \), we solve for \( k \). This gives \( k = 7 \).
Step 4: Conclusion.
Thus, the value of \( k \) is 7.