Question

Matrix A =\[\begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix}\] and matrix B = \[ \begin{bmatrix} 5 & 5 \\ 5 & 5 \end{bmatrix}\]

Statement (1): AB = 0.

Statement (2): AB = 0, even if A ≠ 0 and B ≠ 0

पर्याय

• Both the statements are true.

• Both the statements are false.

• Statement 1 is true and statement 2 is false.

• Statement 1 is false and statement 2 is true.

Answer 1

Both the statements are true.

Explanation:

Given,

Matrix \[A = \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix}\] and matrix \[B = \begin{bmatrix} 5 & 5 \\ 5 & 5 \end{bmatrix}\]

Solving,

⇒ \[AB = \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix} \cdot \begin{bmatrix} 5 & 5 \\ 5 & 5 \end{bmatrix}\]

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

\[{} = \begin{bmatrix} 10 - 10 & 10 - 10 \\ -10 + 10 & -10 + 10 \end{bmatrix}\]

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

So, AB = 0 and also from above example it is proved that AB can be equal to zero even if A ≠ 0 and B ≠ 0.

∴ Both the statements are true.