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  • Show by an example that for A ≠ O, B ≠ O, AB = O.

Show by an example that for A ≠ O, B ≠ O, AB = O.

Answers (1)

We know,

To multiply the given matrices A and B, the number of columns in A must equal the number of rows in B. Thus, if A is an m x n matrix and B is an r x s matrix, n = r.

We are given that,

A ≠ 0 and B ≠ 0

We need to show that, AB = 0.

For multiplication of A and B,

Number of columns of matrix A = Number of rows of matrix B = 2 (let)

Matrices A and B are square matrices of order 2 × 2.

For AB to become 0, one of the column of matrix A and other row of matrix B must be 0.

For example,

\begin{aligned} &A=\left[\begin{array}{ll} 0 & 1 \\ 0 & 4 \end{array}\right]\\ &B=\left[\begin{array}{cc} 3 & -1 \\ 0 & 0 \end{array}\right]\\ &\text { Check: Multiply AB. }\\ &A B=\left[\begin{array}{ll} 0 & 1 \\ 0 & 4 \end{array}\right]\left[\begin{array}{cc} 3 & -1 \\ 0 & 0 \end{array}\right] \end{aligned}

Multiply 1st row of matrix A by matching members of 1st column of matrix B, then finally end by summing them up.

\\(0, 1).(3, 0) = (0 \times 3) + (1 \times 0) \\ \Rightarrow (0, 1).(3, 0) = 0 + 0 = 0

\left[\begin{array}{ll}0 & 1 \\ 0 & 4\end{array}\right]\left[\begin{array}{cc}3 & -1 \\ 0 & 0\end{array}\right]=\left[\begin{array}{ll}0 & \end{array}\right]\\\\ \text{Similarly, let us do it for the rest of the elements.}\\\\ \left[\begin{array}{cc}0 & 1 \\ 0 & 4\end{array}\right]\left[\begin{array}{cc}3 & -1 \\ 0 & 0\end{array}\right]=\left[\begin{array}{cc}0 & (0 \times-1)+(1 \times 0) \\ (0 \times 3)+(4 \times 0) & (0 \times-1)+(4 \times 0)\end{array}\right]\\\\ \Rightarrow\left[\begin{array}{ll}0 & 1 \\ 0 & 4\end{array}\right]\left[\begin{array}{cc}3 & -1 \\ 0 & 0\end{array}\right]=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]\\\\ Hence \ proved.

 

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