Question:
If $A$ is a skew symmetric matrix and $n$ is an even positive integer, then $A^n$ is a
If $A$ is a skew symmetric matrix and $n$ is an even positive integer, then $A^n$ is a
Updated On: Jul 6, 2022
- symmetric matrix
- skew-symmetric matrix
- diagonal matrix
- none of these.
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The Correct Option is A
Solution and Explanation
Given $A' = -A \Rightarrow \left(A'\right)^{n} = \left(-A\right)^{n}$
$ \Rightarrow \left(A^{n}\right)' = \left(\left(-1\right)A\right)^{n} =\left(-1\right)^{n}A^{n} $
$ \Rightarrow \left(A^{n}\right)' = A^{n}$ ($\because $ n is even)
$ \Rightarrow A^{n} $ is a symmetric matrix.
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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.