Question

If the matrix AB is a zero matrix, then which of the following is correct?
A) A must be equal to zero matrix or B must be equal to zero matrix
B) A must be equal to zero matrix and B must be equal to zero matrix
C) It is not necessary that either A is zero matrix or B is zero matrix
D) None of the above

Answer 1

To solve this question, we will start with considering two non-zero matrices. i.e., \left( {\begin{array}{*{20}{c}} 0&0 \\\ 0&1 \end{array}} \right) and \left( {\begin{array}{*{20}{c}} 0&1 \\\ 0&0 \end{array}} \right) .On multiplying these matrices we will check whether we get the product as zero matrix or not. Hence, on getting the product of these matrices, we will get our required answer.

Complete step by step solution:
We have been given that matrix AB is a zero matrix and we are supposed to check whether A and B should be zero matrices or not out of given options mentioned.
So, we will start with considering two non-zero matrices, i.e., A $=$ \left( {\begin{array}{*{20}{c}} 0&0 \\\ 0&1 \end{array}} \right) and B $=$ \left( {\begin{array}{*{20}{c}} 0&1 \\\ 0&0 \end{array}} \right)
Now, on multiplying, the two matrices A $=$ \left( {\begin{array}{*{20}{c}} 0&0 \\\ 0&1 \end{array}} \right) and B $=$ \left( {\begin{array}{*{20}{c}} 0&1 \\\ 0&0 \end{array}} \right), we get
\left( {\begin{array}{*{20}{c}} 0&0 \\\ 0&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 0&1 \\\ 0&0 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {0 \times 0 + 0 \times 0}&{0 \times 1 + 0 \times 0} \\\ {0 \times 0 + 1 \times 0}&{0 \times 1 + 1 \times 0} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0&0 \\\ 0&0 \end{array}} \right)

After multiplying two non-zero matrices, we get a zero matrix.
Hence, to get the answer, we had considered two non-zero matrices and we got the product as a zero matrix.

Thus, option (C), it is not necessary that either A is zero matrix or B is zero matrix, is correct.

Note: In the question above, we have multiplied two matrices. Let us see an example to get a better understanding of matrix multiplication.
\left( {\begin{array}{*{20}{c}} a&b; \\\ c&d; \end{array}} \right)\left( {\begin{array}{*{20}{c}} e&f; \\\ g&h; \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {ae + bg}&{af + bh} \\\ {ce + dg}&{cf + dh} \end{array}} \right)
where, A = \left( {\begin{array}{*{20}{c}} a&b; \\\ c&d; \end{array}} \right) and B = \left( {\begin{array}{*{20}{c}} e&f; \\\ g&h; \end{array}} \right) are two matrices.