Question:
If the matrix AB = O, then
If the matrix AB = O, then
Updated On: Jul 6, 2022
- A = O or B = O
- A = O and B = O
- It is not necessary that either A = O or B = O
- $A \neq O, B \neq O$
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The Correct Option is C
Solution and Explanation
$AB = O \Rightarrow \left|AB \right| = 0 \Rightarrow \left|A\right|. \left|B\right|=0$
$ \Rightarrow \left|A\right| =0 \,or \,\left|B\right| = 0 $
when AB = O, neither A nor B may be O.
For example if
$ A = \begin{bmatrix}1&0\\ 0&0\end{bmatrix} $ and $B = \begin{bmatrix}0&0\\ 1&0\end{bmatrix} $ , then
$AB = \begin{bmatrix}1&0\\ 0&0\end{bmatrix} \begin{bmatrix}0&0\\ 1&0\end{bmatrix} = \begin{bmatrix}0&0\\ 0&0\end{bmatrix} $
But none of A and B are zero matrices.
So if AB is zero it is not necessary that either A = O or B =O.
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Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
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- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.