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Provide Solution for R.D.Sharma Maths Class 12 Chapter 4 Algebra of Matrices Exercise Multiple Choice Questions Question 46 Maths Textbook Solution.

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Answer: Skew symmetric matrix

Given: A and B are symmetric matrices of same order.

Hint: You must know about properties of transpose of a matrix


Given that A and B are symmetric matrices

\begin{aligned} &\Rightarrow A^{T}=A \text { and } B^{T}=B\\ &\text { Now }\\ &\left(A B^{T}-B A^{T}\right)^{T}=\left(A B^{T}\right)^{T}-\left(B A^{T}\right)^{T} \quad\left[\because(A-B)^{T}=A^{T}-B^{T}\right]\\ &=\left(B^{T}\right)^{T} A^{T}-\left(A^{T}\right)^{T} B^{T} \quad\left[\because(A B)^{T}=B^{T} A^{T}\right]\\ &=B A^{T}-A B^{T} \quad\left[\because\left(A^{T}\right)^{T}=A\right] \end{aligned}

\begin{aligned} &=-\left(A B^{T}-B A^{T}\right)\\ &\Rightarrow A B^{T}-B A^{T} \text { Is a skew symmetric matrix } \end{aligned}

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