With reference to the matrix \( A=\begin{bmatrix}5 & 0 & 0\\0 & 5 & 0\\0 & 0 & -5\end{bmatrix} \), which of the following are TRUE?
A. \( |\mathrm{A}|=125 \)
B. A is a scalar matrix.
C. A is a diagonal matrix.
D. \( |\mathrm{A}|=-125 \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → C and D only
The determinant of a diagonal matrix is the product of its diagonal entries:
\( \displaystyle |A|=5\times 5\times(-5)=-125 \)
So \( |A|=-125 \), and A is a diagonal matrix (every off-diagonal entry is 0) but not a scalar matrix (its diagonal entries 5, 5 and −5 are not all equal).
Checking the statements
(A) \( |\mathrm{A}|=125 \) — Incorrect. The entry −5 makes the product −125.
(B) A is a scalar matrix. — Incorrect. A scalar matrix needs all diagonal entries equal; here they are 5, 5 and −5.
(C) A is a diagonal matrix. — Correct. All entries off the diagonal are 0.
(D) \( |\mathrm{A}|=-125 \) — Correct. 5 × 5 × (−5) = −125.