Question:
If $P$ and $Q$ are symmetric matrices of the same order then $PQ - QP$ is
If $P$ and $Q$ are symmetric matrices of the same order then $PQ - QP$ is
Updated On: May 14, 2024
- zero matrix
- identity matrix
- skew symmetric matrix
- symmetric matrix
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The Correct Option is C
Solution and Explanation
Given $P = P'$ and $Q=Q'$
$\left(PQ-QP\right)'=\left(PQ\right)'-\left(QP\right)^{'}$
$=\left(Q'P'- P'Q'\right)=QP-PQ$
$-\left[PQ-QP\right]$
$\therefore\, \left(PQ-QP\right)$ is skew symmetric.
$\left(PQ-QP\right)'=\left(PQ\right)'-\left(QP\right)^{'}$
$=\left(Q'P'- P'Q'\right)=QP-PQ$
$-\left[PQ-QP\right]$
$\therefore\, \left(PQ-QP\right)$ is skew symmetric.
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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.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.