If $ A=\left[ \begin{matrix} 3 & -4 \\ 1 & -1 \\ \end{matrix} \right], $ then $ (A-A') $ is equal to (where, $A'$ is transpose of matrix $A$)
- null matrix
- identity matrix
- symmetric
- skew-symmetric
The Correct Option is D
Solution and Explanation
The correct option is(D): skew-symmetric.
Given, \(A=\left[ \begin{matrix} 3 & -4 \\ 1 & -1 \\ \end{matrix} \right]\)
Then, \(A'=\left[ \begin{matrix} 3 & 1 \\ -4 & -1 \\ \end{matrix} \right]\)
Now, \(A-A'=\left[ \begin{matrix} 3 & -4 \\ 1 & -1 \\ \end{matrix} \right]-\left[ \begin{matrix} 3 & 1 \\ -4 & -1 \\ \end{matrix} \right]\)
\(\Rightarrow\) \(A-A'=\left[ \begin{matrix} 0 & -5 \\ 5 & 0 \\ \end{matrix} \right]\) .. (i)
Now, we have \(A'-A=\left[ \begin{matrix} 3 & 1 \\ -4 & -1 \\ \end{matrix} \right]-\left[ \begin{matrix} 3 & -4 \\ 1 & -1 \\ \end{matrix} \right]=\left[ \begin{matrix} 0 & 5 \\ -5 & 0 \\ \end{matrix} \right]\)
\(\Rightarrow\) \((A'-A)'=\left[ \begin{matrix} 0 & -5 \\ 5 & 0 \\ \end{matrix} \right]=(A-A')\)
[From E (i)] which represent that \((A-A')\) is skew-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.