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Explain Solution RD Sharma Class 12 Chapter Algebra of Matrices Exercise 4.5 Question 1 Maths.

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Answer: $A - A^{T}$ is a skew-symmetric matrix.

Hint: Matrix $B$ is said to be skew-symmetric if transpose of matrix $A$ is equal to negative of matrix $A$ i.e $(B^{T} = -B)$

Find A^T and prove that $(A -A^{T}) = - (A - A^{T})$

Given: $A =\begin{bmatrix} 2 &3 \\ 4& 5 \end{bmatrix}$

Solution:

$A^T= \begin{bmatrix} 2 & 4\\ 3 &5 \end{bmatrix}$

$A - A^{T} = \begin{bmatrix} 2 &3 \\ 4& 5 \end{bmatrix}-\begin{bmatrix} 2 &4 \\ 3& 5 \end{bmatrix}$

$= \begin{bmatrix} 0 &-1 \\ 1& 0 \end{bmatrix}$

$= B$

$B^{T} = \begin{bmatrix} 0 &1 \\ -1 &0 \end{bmatrix}$

$-B^{T} = \begin{bmatrix} 0 &-1 \\ 1 &0 \end{bmatrix}$

$- B^{T}= B$

Hence,

$B$ is a skew-symmetric matrix.

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