Question:
if $A = \begin{bmatrix}3&-4\\ 1&-1\end{bmatrix}$ is the sum of a symmetric matrix $B$ and a skew-symmetric matrix $C$, then $C$ is
if $A = \begin{bmatrix}3&-4\\ 1&-1\end{bmatrix}$ is the sum of a symmetric matrix $B$ and a skew-symmetric matrix $C$, then $C$ is
Updated On: Jul 6, 2022
- $\begin{bmatrix}1&-\frac{5}{2}\\ \frac{5}{2}&0\end{bmatrix}$
- $\begin{bmatrix}1&-\frac{5}{2}\\ \frac{5}{2}&1\end{bmatrix}$
- $\begin{bmatrix}0&-\frac{5}{2}\\ \frac{5}{2}&0\end{bmatrix}$
- $\begin{bmatrix}0&-\frac{3}{2}\\ \frac{5}{2}&1\end{bmatrix}$
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The Correct Option is C
Solution and Explanation
$A = \begin{bmatrix}3&-4\\ 1&-1\end{bmatrix} $
$A= \left(\frac{A + A'}{2}\right)+\left(\frac{A-A'}{2}\right) = B + C$
[where $B$ and $C$ are symmetric and skew-symmetric matrices respectively]
Now, $C = \frac{A - A'}{2} = \frac{1}{2 } \left\{\begin{bmatrix}3&-4\\ 1&-1\end{bmatrix} - \begin{bmatrix}3&1\\ -4&-1\end{bmatrix}\right\}$
$ = \frac{1}{2}\begin{bmatrix}0&-5\\ 5&0\end{bmatrix} = \begin{bmatrix}0&-\frac{5}{2}\\ \frac{5}{2}&0\end{bmatrix}$
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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.