If A = \(\left[\begin{array}{cc}2 & −3 \\3 & 5\end{array}\right]\), 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.

This question was previously asked in
CUET Mathematics 30th Aug 2022 Official Paper
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  1. A, B, C only
  2. A, D, E only
  3. A, B, D only
  4. C, D, E only

Answer (Detailed Solution Below)

Option 3 : A, B, D only
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A. In the above matrix A = \(\left[\begin{array}{cc}2 & −3 \\3 & 5\end{array}\right]\), we can see, the number of rows and columns are 2 respectively. Since the order of the matrix is 2 × 2, hence A is a square matrix.

B. The given 2 × 2 matrix \(A = \left[ {\begin{array}{*{20}{c}} 2&-3\\ 3&5 \end{array}} \right]\)

We first find the determinant of A.

Det A = (2 × 5) - (3 × -3) = 10 + 9 = 19

∴ |A| = 19

Since, |A| ≠ 0 ⇒ A-1 exists.

C. To know if a matrix is symmetric, find the transpose of that matrix. If the transpose of that matrix is equal to itself, it is a symmetric matrix. That is A = AT

Here A = \(\left[\begin{array}{cc}2 & −3 \\3 & 5\end{array}\right]\) then AT = \(\left[\begin{array}{cc}2 & 3 \\-3 & 5\end{array}\right]\)

Here, A ≠  AT

Thus A is not a symmetric matrix. 

D. We have already derived |A| 19.

E. Null MatrixIf in a matrix all the elements are zero then it is called a null matrix. It is also called a zero matrix. Here we can see A is not null matrix. 

Thus A, B, D is the correct answer. 

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