For the matrix \( A=\begin{bmatrix}0 & 0 & 5\\ 0 & 6 & 0\\ 7 & 0 & 0\end{bmatrix} \), which of the following statement(s) is/are TRUE?
A. \( A \) is a scalar matrix.
B. \( A \) is a diagonal matrix.
C. \( A \) is a square matrix.
D. \( A \) is a skew symmetric matrix.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → C only
A has 3 rows and 3 columns, and its only non-zero entries are \( a_{13}=5 \), \( a_{22}=6 \) and \( a_{31}=7 \).
So A is a square matrix with non-zero entries off the leading diagonal (\( a_{13}=5 \), \( a_{31}=7 \)) and a non-zero diagonal entry \( a_{22}=6 \).
Checking the statements
(A) A is a scalar matrix — Incorrect. A scalar matrix is diagonal with equal diagonal entries, but \( a_{13}=5\ne0 \).
(B) A is a diagonal matrix — Incorrect. \( a_{13}=5 \) and \( a_{31}=7 \) are non-zero and lie off the leading diagonal.
(C) A is a square matrix — Correct. It has 3 rows and 3 columns.
(D) A is a skew symmetric matrix — Incorrect. A skew-symmetric matrix has zero diagonal, but \( a_{22}=6 \); also \( a_{31}=7\ne-a_{13}=-5 \).