JEE MAIN - Mathematics (2022 - 28th July Evening Shift - No. 3)
Let $$\mathrm{A}$$ and $$\mathrm{B}$$ be any two $$3 \times 3$$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?
$$\mathrm{A}^{4}-\mathrm{B}^{4}$$ is a smmetric matrix
$$\mathrm{AB}-\mathrm{BA}$$ is a symmetric matrix
$$\mathrm{B}^{5}-\mathrm{A}^{5}$$ is a skew-symmetric matrix
$$\mathrm{AB}+\mathrm{BA}$$ is a skew-symmetric matrix
Explanation
(A) $$M = {A^4} - {B^4}$$
$${M^T} = {({A^4} - {B^4})^T} = {({A^T})^4} - {({B^T})^4}$$
$$ = {A^4} - {( - B)^4} = {A^4} - {B^4} = M$$
(B) $$M = AB - BA$$
$${M^T} = {(AB - BA)^T} = {(AB)^T} - {(BA)^T}$$
$$ = {B^T}{A^T} - {A^T}{B^T}$$
$$ = - BA - A( - B)$$
$$ = AB - BA = M$$
(C) $$M = {B^5} - {A^5}$$
$${M^T} = {({B^T})^5} - {({A^T})^5} = - ({B^5} + {A^5}) \ne - M$$
(D) $$M = AB + BA$$
$${M^T} = {(AB)^T} + {(BA)^T}$$
$$ = {B^T}{A^T} + {A^T}{B^T} = - BA - AB = - M$$
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