Question

AB=0, if and only if
A . $A\ne 0,B=0$
B . $A=0,B\ne 0$
C . $A=0,B=0$
D . None of these

Answer 1

Hint: To find out which option follows, first we consider the two non- zero matrices A and B.
Matrix are the set of numbers which are arranged in rows and columns to make a rectangular array. In matrix, the numbers are called the entries or the entities of the matrix. Then we multiply A and B matrices and check whether the product gives zero matrix or not. Hence on getting the product of the matrices, we are able to get our correct option.

Complete Step- by- step Solution:
Given $AB=0$
Consider two non- zero matrices A = $\left[\begin{matrix}
   0 & 0 \\
   0 & 1 \\
\end{matrix} \right]$ and B = $\left[ \begin{matrix}
   0 & 1 \\
   0 & 0 \\
\end{matrix} \right]$
We see that the both matrices are of $2\times 2$order.
Now we multiply the matrix A with B, we get
AB = $\left[ \begin{matrix}
   0 & 0 \\
   0 & 1 \\
\end{matrix} \right]$ $\left[ \begin{matrix}
   0 & 1 \\
   0 & 0 \\
\end{matrix} \right]$
AB = $\left[ \begin{matrix}
   0\times 0+0\times 0 & 0\times 1+0\times 0 \\
   0\times 0+1\times 0 & 0\times 1+1\times 0 \\
\end{matrix} \right]$
Now on simplifying the above equation, we get
AB = $\left[ \begin{matrix}
   0 & 0 \\
   0 & 0 \\
\end{matrix} \right]$
Thus, we see that multiplication of two non- zero matrices gives a zero matrix.
Then $A\ne 0,B\ne 0$

Thus, Option ( D) is correct.

Note: In these type of questions, Students take care while multiplying the two matrices. In multiplying the matrices, element of row 1 multiplied with the element of column 1 of another matrix and then we multiply the second element of row 1 with the second element of column 1 of the other matrix. To multiply the two matrices, remember that the number of column of the first matrix match the number of rows of the second matrix.