Question

Two matrices \[A\] and \[B\] each of order \[2 \times 2.\]

Assertion (A): \[A \times B = 0 \nRightarrow A = 0\] or \[B = 0.\]

Reason (R): Let \[A = \begin{bmatrix} 2 & 2 \\ 5 & 5 \end{bmatrix} \neq 0\] and \[B = \begin{bmatrix} -4 & 3 \\ 4 & -3 \end{bmatrix} \neq 0\]

but \[A \times B = \begin{bmatrix} 2 & 2 \\ 5 & 5 \end{bmatrix} \begin{bmatrix} -4 & 3 \\ 4 & -3 \end{bmatrix} = 0\]

Options

• A is true, R is false.

• A is false, R is true.

• Both A and R are true and R is the correct reason for A.

• Both A and R are true and R is the incorrect reason for A.

Answer 1

Both A and R are true and R is the correct reason for A.

Explanation:

It is not necessarily true in case of matrices that if \[A \times B = 0\]

Then, either A = 0 or B = 0.

So, assertion (A) is true.

According to reason, \[A = \begin{bmatrix} 2 & 2 \\ 5 & 5 \end{bmatrix}\] and \[B = \begin{bmatrix} -4 & 3 \\ 4 & -3 \end{bmatrix}\]

AB = \[A = \begin{bmatrix} 2 & 2 \\ 5 & 5 \end{bmatrix}\] . \[ \begin{bmatrix} -4 & 3 \\ 4 & -3 \end{bmatrix}\]

\[{} = \begin{bmatrix} 2 \times (-4) + 2 \times 4 & 2 \times 3 + 2 \times (-3) & 5 \times (-4) + 5 \times 4 & 5 \times 3 + 5 \times (-3) \end{bmatrix}\]

\[{} = \begin{bmatrix} -8 + 8 & 6 - 6 & -20 + 20 & 15 - 15 \end{bmatrix}\]

\[{} = \begin{bmatrix} 0 & 0 & 0 & 0 \end{bmatrix}\]

So, reason (R) is true.

Thus, Both A and R are true and R is correct reason for A.