If A is a square matrix […] such that A^2=pA , then the value of p is,

The correct answer is Option (3) → 10

Squaring A:

    \( \displaystyle A^{2}=\begin{bmatrix}5&-5\\-5&5\end{bmatrix}\begin{bmatrix}5&-5\\-5&5\end{bmatrix} \)

    \( \displaystyle =\begin{bmatrix}25+25&-25-25\\-25-25&25+25\end{bmatrix} \)

    \( \displaystyle =\begin{bmatrix}50&-50\\-50&50\end{bmatrix} \)

    \( \displaystyle =10A \)

So p = 10.

Shortcut: By the Cayley-Hamilton theorem for a 2 × 2 matrix,

    \( \displaystyle A^{2}-(\mathrm{tr}\,A)A+|A|I=O \)

Here \( \mathrm{tr}\,A=10 \) and \( |A|=25-25=0 \), so \( A^{2}=10A \) at once.