Question

If $A = \begin{bmatrix} 1 & 0 & 0 \\ a & -1 & 0 \\ b & c & 1 \end{bmatrix}$ is such that $A^2 = I$, then

• A. $b = \dfrac{ac}{2}$
• B. $b = -\dfrac{ac}{2}$
• C. $b = \dfrac{a + c}{2}$
• D. $b = \sqrt{ac}$

Hint:

When verifying $A^2 = I$, matrix multiplication and equating corresponding entries is a key technique.

Answer 1

The Correct Option is B

Given $A^2 = I$, compute $A^2$ using the given matrix.
Multiply $A$ with itself and equate with the identity matrix.
Comparing the third row, first column entry: $b + ac + b = 0 \Rightarrow 2b + ac = 0 \Rightarrow b = -\dfrac{ac}{2}$