If matrix $A =\begin{bmatrix}p&-3\\-4&p\end{bmatrix}$ and $|A^3|=64$, then the value of $p$ is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → ±4
$A = \begin{bmatrix} p & -3 \\ -4 & p \end{bmatrix}$
$|A^3| = |A|^3 = 64$
$|A| = p \cdot p - (-3)(-4) = p^2 - 12$
$(p^2 - 12)^3 = 64$
$p^2 - 12 = 4 \quad \text{or} \quad p^2 - 12 = -4$
From $p^2 - 12 = 4$: $p^2 = 16 \ \Rightarrow \ p = \pm 4$
From $p^2 - 12 = -4$: $p^2 = 8 \ \Rightarrow \ p = \pm 2\sqrt{2}$
Values of $p$: $\pm 4, \ \pm 2\sqrt{2}$