
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
164.4k+ views
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.
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.
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