Question:
If $A = \begin{bmatrix}6&8&5\\ 4&2&3\\ 9&7&1\end{bmatrix} $ is the sum of a symmetric matrix B and skew-symmetric matrix C, then B is
If $A = \begin{bmatrix}6&8&5\\ 4&2&3\\ 9&7&1\end{bmatrix} $ is the sum of a symmetric matrix B and skew-symmetric matrix C, then B is
Updated On: Jul 6, 2022
- $ \begin{bmatrix}6&6&7\\ 6&2&5\\ 7&5&1\end{bmatrix}$
- $\begin{bmatrix}0&2&-2\\ -2&5&-2\\ 2&2&0\end{bmatrix}$
- $\begin{bmatrix}6&6&7\\ -6&2&-5\\ -7&5&1\end{bmatrix}$
- $\begin{bmatrix}0&6&-2\\ 2&0&-2\\ -2&-2&0\end{bmatrix}$
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The Correct Option is A
Solution and Explanation
If $A = \begin{bmatrix}6&8&5\\ 4&2&3\\ 9&7&1\end{bmatrix} $ is the sum of a symmetric
matrix B and skew symmetric matrix C,
Transpose of $A = \begin{bmatrix}6&4&9\\ 8&2&7\\ 5&3&1\end{bmatrix}$
So that $B = \frac{1}{2} \left[\begin{bmatrix}6&8&5\\ 4&2&3\\ 9&7&1\end{bmatrix} + \begin{bmatrix}6&4&9\\ 8&2&7\\ 5&3&1\end{bmatrix}\right] $
$B = \frac{1}{2} \begin{bmatrix}12&12&14\\ 12&4&10\\ 14&10&2\end{bmatrix} = \begin{bmatrix}6&6&7\\ 6&2&5\\ 7&5&1\end{bmatrix}$
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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.