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Let $X$ and $Y$ be two arbitrary, $3 \times 3$, non-zero, skew-symmetric matrices and $Z$ be an arbitrary $3 \times 3$, nonzero, symmetric matrix. Then which of the following matrices is (are) skew symmetric ?
Let $X$ and $Y$ be two arbitrary, $3 \times 3$, non-zero, skew-symmetric matrices and $Z$ be an arbitrary $3 \times 3$, nonzero, symmetric matrix. Then which of the following matrices is (are) skew symmetric ?
Updated On: Jun 14, 2022
- $Y^3Z^4 - Z^4Y^3$
- $X^{44} + Y^{44}$
- $X^4Z^3 - Z^3X^4$
- $X^{23} + Y^{23}$
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The Correct Option is D
Solution and Explanation
$X^{T} = -X Y^{T} = -Y Z^{T} = Z$
$\left(A\right) \left(Y^{3} Z^{4} - Z^{4} Y^{3}\right)^{T} = \left(Y^{3} Z^{4}\right)^{T }- \left(Z^{4} Y^{3}\right)^{T}$
$= \left(Z^{4}\right)^{T} \left(Y^{3}\right)^{T }- \left(Y^{3}\right)^{T} \left(Z^{4}\right)^{T}$
$= \left(Z^{T}\right)^{4} \left(Y^{T}\right)^{3} - \left(Y^{T}\right)^{3} \left(Z^{T}\right)^{4}$
$= Z^{4 }\left(-Y\right)^{3} - \left(-Y\right)^{3} \left(Z\right)^{4}$
$= -Z^{4}Y^{3} + Y^{3} Z^{4}$
$= Y^{3}Z^{4} - Z^{4}Y^{3}$
Hence it is symmetric matrix.
$\left(B\right) \left(X^{44} + Y^{44}\right)^{T} = \left(X^{T}\right)^{44} + \left(Y^{T}\right)^{44}$
$= X^{44} + Y^{44}$
Hence it is symmetric matrix.
$\left(C\right) \left(X^{4} Z^{3} - Z^{3} X^{4}\right)^{T} = \left(X^{4} Z^{3}\right)^{T} - \left(Z^{3} X^{4}\right)^{T}$
$= \left(Z^{T}\right)^{3} \left(X^{T}\right)^{4} - \left(X^{T}\right)^{4} \left(Z^{T}\right)^{3}$
$= Z^{3} X^{4} - X^{4 }Z^{3}$
$= - \left(X^{4 }Z^{3} - Z^{3} X^{4}\right)$
Hence it is skew symmetric matrix.
$\left(D\right) \left(X^{23} + Y^{23}\right)^{T} = \left(X^{T}\right)^{23} + \left(Y^{T}\right)^{23}$
$= - \left(X^{23} + Y^{23}\right)$
Hence it is skew symmetric matrix.
$\left(A\right) \left(Y^{3} Z^{4} - Z^{4} Y^{3}\right)^{T} = \left(Y^{3} Z^{4}\right)^{T }- \left(Z^{4} Y^{3}\right)^{T}$
$= \left(Z^{4}\right)^{T} \left(Y^{3}\right)^{T }- \left(Y^{3}\right)^{T} \left(Z^{4}\right)^{T}$
$= \left(Z^{T}\right)^{4} \left(Y^{T}\right)^{3} - \left(Y^{T}\right)^{3} \left(Z^{T}\right)^{4}$
$= Z^{4 }\left(-Y\right)^{3} - \left(-Y\right)^{3} \left(Z\right)^{4}$
$= -Z^{4}Y^{3} + Y^{3} Z^{4}$
$= Y^{3}Z^{4} - Z^{4}Y^{3}$
Hence it is symmetric matrix.
$\left(B\right) \left(X^{44} + Y^{44}\right)^{T} = \left(X^{T}\right)^{44} + \left(Y^{T}\right)^{44}$
$= X^{44} + Y^{44}$
Hence it is symmetric matrix.
$\left(C\right) \left(X^{4} Z^{3} - Z^{3} X^{4}\right)^{T} = \left(X^{4} Z^{3}\right)^{T} - \left(Z^{3} X^{4}\right)^{T}$
$= \left(Z^{T}\right)^{3} \left(X^{T}\right)^{4} - \left(X^{T}\right)^{4} \left(Z^{T}\right)^{3}$
$= Z^{3} X^{4} - X^{4 }Z^{3}$
$= - \left(X^{4 }Z^{3} - Z^{3} X^{4}\right)$
Hence it is skew symmetric matrix.
$\left(D\right) \left(X^{23} + Y^{23}\right)^{T} = \left(X^{T}\right)^{23} + \left(Y^{T}\right)^{23}$
$= - \left(X^{23} + Y^{23}\right)$
Hence it is skew symmetric matrix.
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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.
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