If \(A = \begin{bmatrix}2 & -3\\3 & 5\end{bmatrix}\), then which of the following statements are correct?
A. A is a square matrix
B. A−1 exists
C. A is a symmetric matrix
D. |A| = 19
E. A is a null matrix
Choose the correct answer from the options given below.
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → A, B, D only
A has 2 rows and 2 columns. Its determinant:
\(|A| = 2(5) - (-3)(3)\)
\(= 10 + 9 = 19\)
Its transpose:
\( \displaystyle A^{T} = \begin{bmatrix}2 & 3\\-3 & 5\end{bmatrix} \ne A \)
So A is a 2 × 2 matrix with |A| = 19 ≠ 0, \(A^{T} \ne A\), and its entries are not all zero.
Checking the statements
(A) A is a square matrix — Correct. It has 2 rows and 2 columns.
(B) A−1 exists — Correct. |A| = 19 ≠ 0, so A is non-singular.
(C) A is a symmetric matrix — Incorrect. \(a_{12} = -3\) but \(a_{21} = 3\).
(D) |A| = 19 — Correct. |A| = 10 + 9 = 19.
(E) A is a null matrix — Incorrect. Its entries are not all zero.