Matrices — Assertion-Reason Type Questions

A, B and C are square matrices of order 2 such that AB = C.

Assertion (A): BA = C

Reason (R): Matrix multiplication is not always commutative.

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Given, A, B and C are square matrices of order 2 such that AB = C.

Matrix multiplication is not always commutative.

So, reason (R) is true.

⇒ AB ≠ BA

So, BA is not necessarily equal to C.

So, assertion (A) is false.

Thus, Assertion (A) is false, but Reason (R) is true.

Hence, option 2 is the correct option.

A, B and C are square matrices of order 2 such that AB = C.

Assertion (A): Product BA need not be equal to C.

Reason (R): Matrix multiplication is not associative.

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Given, A, B and C are square matrices of order 2 such that AB = C.

Matrix multiplication is not always commutative.

∴ AB is not necessarily equal to BA.

So, BA is not necessarily equal to C.

So, assertion (A) is true.

Matrix multiplication is associative.

So, reason (R) is false.

Assertion (A) is true, but Reason (R) is false.

Hence, option 1 is the correct option.

If A = $\begin{bmatrix*}[r] 3 & -2 \end{bmatrix*} \text{ and B} = \begin{bmatrix*}[r] -1 & 4 \\ 2 & 0 \end{bmatrix*}$

Assertion (A): Product AB of the two matrices A and B is possible.

Reason (R): Number of columns of matrix A is equal to number of rows in matrix B.

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Given, A = $\begin{bmatrix*}[r] 3 & -2 \end{bmatrix*} \text{ and B} = \begin{bmatrix*}[r] -1 & 4 \\ 2 & 0 \end{bmatrix*}$

To multiply two matrices, the number of columns of the first matrix must equal the number of rows of the second matrix.

Matrix A has 1 row and 2 columns.

Matrix B has 2 rows and 2 columns.

So, reason (R) is true.

The number of columns in A is 2 and number of rows in B is also 2.

So, product AB is defined.

So, both A and R are true and R is the correct explanation of assertion A.

Thus, both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

Hence, option 3 is the correct option.