The determinant of which of the following matrices is a perfect square?
A. \( \begin{bmatrix}5 & 29\\-1 & 6\end{bmatrix} \)
B. \( \begin{bmatrix}5 & 26\\1 & 6\end{bmatrix} \)
C. \( \begin{bmatrix}10 & 3\\17 & 10\end{bmatrix} \)
D. \( \begin{bmatrix}8 & 5\\-3 & -8\end{bmatrix} \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → B and C only
For a 2 × 2 matrix, the determinant is \( ad-bc \):
\( \displaystyle \text{A}:\ 5(6)-29(-1)=30+29=59 \)
\( \displaystyle \text{B}:\ 5(6)-26(1)=30-26=4 \)
\( \displaystyle \text{C}:\ 10(10)-3(17)=100-51=49 \)
\( \displaystyle \text{D}:\ 8(-8)-5(-3)=-64+15=-49 \)
So the determinants are 59, 4, 49 and −49.
Checking the statements
(A) \( \begin{bmatrix}5 & 29\\-1 & 6\end{bmatrix} \) — Incorrect. 59 is not a perfect square.
(B) \( \begin{bmatrix}5 & 26\\1 & 6\end{bmatrix} \) — Correct. \( 4=2^2 \).
(C) \( \begin{bmatrix}10 & 3\\17 & 10\end{bmatrix} \) — Correct. \( 49=7^2 \).
(D) \( \begin{bmatrix}8 & 5\\-3 & -8\end{bmatrix} \) — Incorrect. −49 is negative, so it cannot be a perfect square.