Algebra

Matrices & Determinants

Medium

common

If $A = \begin{bmatrix} 2 & 1 & 3 \\ 4 & -3 & 5 \end{bmatrix}$ and $B = \begin{bmatrix} -2 & 3 \\ 4 & -5 \\ 1 & 2 \end{bmatrix}$ , then which of the following statements are TRUE?

(A) AB is defined

(B) AB and BA both are defined and AB = I, where I is an identity matrix of order 2

(C) BA is defined

(D) AB and BA both are defined and AB = BA

Choose the correct answer from the options given below:

(A), (B) and (C) only

(B), (C) and (D) only

(A) and (C) only

(A), (C) and (D) only

✅ Correct Option: 3

Step-by-Step Explanation:

For matrix multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix.

Check the order (dimensions) of the matrices:

Matrix $A$ has $2$ rows and $3$ columns $\implies$ Order of $A = 2 \times 3$
Matrix $B$ has $3$ rows and $2$ columns $\implies$ Order of $B = 3 \times 2$

Evaluate Statement (A): Is $AB$ defined?

We are multiplying a $(2 \times \mathbf{3})$ matrix by a $(\mathbf{3} \times 2)$ matrix.
Since the inner dimensions match ( $3 = 3$ ), $AB$ is defined. The resulting matrix will have an order of $2 \times 2$ .

Evaluate Statement (C): Is $BA$ defined?

We are multiplying a $(3 \times \mathbf{2})$ matrix by a $(\mathbf{2} \times 3)$ matrix.
Since the inner dimensions match ( $2 = 2$ ), $BA$ is defined. The resulting matrix will have an order of $3 \times 3$ .

Evaluate Statements (B) and (D):

Since the resulting matrix $AB$ is of order $2 \times 2$ and $BA$ is of order $3 \times 3$ , they cannot be equal to each other ( $AB \neq BA$ ).
Therefore, statements (B) and (D) are false.

Thus, only statements (A) and (C) are true.

2026: 18 May Shift 1

If $\begin{vmatrix}6 & 7\\ 3 & 8\end{vmatrix} = \begin{vmatrix}3x & 6\\ -1 & x\end{vmatrix}$ , then the sum of values of $x$ is equal

2026: 30 May Shift 2

If $a_{ij}$ and $A_{ij}$ represents the elements and cofactors of the matrix $A = \begin{bmatrix}2 & 1 & 3\\3 & 2 & 1\\1 & 2 & 3\end{bmatrix}$ , then which of the following statements are True?

(A) $|A| = 28$

(B) $A_{23} = -3$

(D) $\frac{A_{21}}{A_{13}} = \frac{3}{4}$

Choose the correct answer from the options given below:

2023: 15 June Shift 2

Match List - I with List - II. If $A = \begin{vmatrix} 3 & -2 & 3 \\ 2 & 1 & -1 \\ 4 & -3 & 2 \end{vmatrix}$

List - I	List - II
(A) $M_{23}$	(I) $-17$
(B) $A_{32} + a_{13}$	(II) $-1$
(C) A	(III) 0
(D) $a_{13}A_{12} + a_{23}A_{22} + a_{33}A_{32}$	(IV) 12

Choose the correct answer from the options given below :

2026: 12 May Shift 2

If $[2\ \ x\ \ 3]\begin{bmatrix}-1 & 0 & -1\\ -1 & 1 & 0\\ 0 & 1 & 1\end{bmatrix}\begin{bmatrix}x\\0\\-1\end{bmatrix}=[-4]$ , then the value(/s) of $x$ is

2026: 19 May Shift 1

If $A = \begin{bmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{bmatrix}$ , then which of the following statements are TRUE?

[where $A_{ij}$ is the cofactor of the (i, j)th element, $a_{ij}$ of matrix A, $1 \gt i, j \le 3$ .]

(A) $a_{11}A_{31} = -24$

(B) $a_{12}A_{32} = -66$

(D) $a_{21}A_{21} = -24$

Choose the correct answer from the options given below:

2025: 21 May Shift 2

If $A = \begin{bmatrix} 4 & 3 \\ 2 & -1 \\ 1 & 0 \end{bmatrix}$ and $B^T = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 1 & -1 \end{bmatrix}$ , then $A - B$ is equal to

Solution

Step-by-Step Explanation:

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