If matrix A and A^2 = 15I, what are the possible values of p?

Question 73

HardOriginal question

If $A = \begin{bmatrix}p & 2 \\ 3 & - p\end{bmatrix}$ and $A^2 = 15I$ , what are the possible values of $p$ ?

p = 3

onlyCorrect answerYour answer

p = \pm\sqrt{15}

Correct answerYour answer

p = \pm 9

Correct answerYour answer

p = \pm 3

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

$A^2 = \begin{bmatrix}p^{2} + 6 & 0 \\ 0 & p^{2} + 6\end{bmatrix}$ , which is $(p^2 + 6)I$ .
$p^2 + 6 = 15$ gives $p^2 = 9$ , so $p = 3$ or $p = -3$ .

Worth noting

$\pm\sqrt{15}$ sets $p^2 = 15$ and forgets the $6$ from $2 \times 3$ .

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