If A = [ 2 - 3 3 5 ] , 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 b

Question

If $A = [\begin{matrix} 2 & - 3 \\ 3 & 5 \end{matrix}],$ 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.

Smart Achievers · Accepted Answer

(3)

Ans. (A), (B), (D) only

Explanation:

(A) $A = {[\begin{matrix} 2 & - 3 \\ 3 & 5 \end{matrix}]}_{2 \times 2}$

(B) $| A |$ $= (2) (5) - (- 3) (3) = 10 + 9 = 19$

$∵$ $| A |$ $\neq 0$

$∴$ $A^{- 1} exists$

(C) For a symmetric matrix,

$A = A^{T}$

but $[\begin{matrix} 2 & - 3 \\ 3 & 5 \end{matrix}] \neq [\begin{matrix} 2 & 3 \\ - 3 & 5 \end{matrix}]$ (False)

(D) $| A |$ $= [\begin{matrix} 2 & - 3 \\ 3 & 5 \end{matrix}] = 19$

(E) The matrix whose every element is zero is called a null matrix. Therefore, A is not a null matrix.

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