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  • Which of the following statements are True or False If matrix AB = O, then A = O or B = O or both A and B are null matrices.

Which of the following statements are True or False
If matrix AB = O, then A = O or B = O or both A and B are null matrices.

Answers (1)

False

Its not necessary that for multiplication of matrix A and B to be 0 one of them has to be a null matrix.
For example,

\begin{array}{c} \text {Let } A=\left[\begin{array}{ccc} 1 & 1 & -1 \\ 1 & 1 & -1 \\ 1 & 1 & -1 \end{array}\right] \text { and } B=\left[\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 2 & 2 & 2 \end{array}\right] \\ A \times B=\left[\begin{array}{ccc} 1 & 1 & -1 \\ 1 & 1 & -1 \\ 1 & 1 & -1 \end{array}\right]\left[\begin{array}{lll} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 2 & 2 & 2 \end{array}\right]=\left[\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \end{array}

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