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# If A and B are symmetric matrices of the same order, then ABT – BAT is a (a) skew-symmetric matrix (b) null matrix (c) symmetric matrix (d) none of these

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Solution

## It is given that, A and B are symmetric matrices of the same order. ∴ AT = A and BT = B .....(1) Now, ${\left(A{B}^{T}-B{A}^{T}\right)}^{T}$ $={\left(AB-BA\right)}^{T}$ [Using (1)] $={\left(AB\right)}^{T}-{\left(BA\right)}^{T}\left[{\left(X+Y\right)}^{T}={X}^{T}+{Y}^{T}\right]$ $={B}^{T}{A}^{T}-{A}^{T}{B}^{T}\left[{\left(XY\right)}^{T}={Y}^{T}{X}^{T}\right]$ $=BA-AB$ [Using (1)] $=-\left(A{B}^{T}-B{A}^{T}\right)$ [Using (1)] We know that, a matrix X is skew-symmetric if ${X}^{T}=-X$. Since ${\left(A{B}^{T}-B{A}^{T}\right)}^{T}=-\left(A{B}^{T}-B{A}^{T}\right)$, therefore, ABT – BAT is a skew-symmetric matrix. Hence, the correct answer is option (a).

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