Question:
The symmetric part of the matrix $A =\begin{bmatrix}
{1}&{2} &{4}\\
{6}&{8}& {2} \\
{2}&{-2}&{7}\\
\end{bmatrix} $ is
The symmetric part of the matrix $A =\begin{bmatrix}
{1}&{2} &{4}\\
{6}&{8}& {2} \\
{2}&{-2}&{7}\\
\end{bmatrix} $ is
Updated On: Feb 23, 2024
- $\begin{bmatrix} {1}&{4} &{3}\\ {2}&{8}& {0} \\ {3}&{0}&{7}\\ \end{bmatrix} $
- $\begin{bmatrix} {1}&{4} &{3}\\ {4}&{8}& {0} \\ {3}&{0}&{7}\\ \end{bmatrix} $
- $\begin{bmatrix} {0}&{-2} &{-1}\\ {-2}&{0}& {-2} \\ {-1}&{-2}&{0}\\ \end{bmatrix} $
- $\begin{bmatrix} {0}&{-2} &{1}\\ {2}&{0}& {2} \\ {-1}&{2}&{0}\\ \end{bmatrix} $
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The Correct Option is B
Solution and Explanation
Given, matrix $A= \begin{bmatrix}1 & 2 & 4 \\ 6 & 8 & 2 \\ 2 & -2 & 7\end{bmatrix}$
$\therefore$ Symmetric part of $A=\frac{1}{2}\left[A+A'\right]$
$=\frac{1}{2}\left\{\begin{bmatrix}1 & 2 & 4 \\ 6 & 8 & 2 \\ 2 & -2 & 7\end{bmatrix}+ \begin{bmatrix}1 & 6 & 2 \\ 2 & 8 & -2 \\ 4 & 2 & 7\end{bmatrix}\right\}$
$=\frac{1}{2}\left\{\begin{bmatrix}2 & 8 & 6 \\ 8 & 16 & 0 \\ 6 & 0 & 14\end{bmatrix}\right\}= \begin{bmatrix}1 & 4 & 3 \\ 4 & 8 & 0 \\ 3 & 0 & 7\end{bmatrix}$
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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.