Verified Solution • 2025–26 Session moderate

QJEEM22Ma6288: JEE Main 2022 Mathematics - Class 12 JEEM22Ma6288 Solution: Let and be any two symmetric and skew symmetric matrices…

Let A\mathrm{A} and B\mathrm{B} be any two 3×33 \times 3 symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

Multiple Choice Options

  • A A4−B4\mathrm{A}^{4}-\mathrm{B}^{4} is a smmetric matrix
  • B AB−BA\mathrm{AB}-\mathrm{BA} is a symmetric matrix
  • C B5−A5\mathrm{B}^{5}-\mathrm{A}^{5} is a skew-symmetric matrix
  • D AB+BA\mathrm{AB}+\mathrm{BA} is a skew-symmetric matrix

Step-by-Step Verified Solution

(A) M=A4−B4M = {A^4} - {B^4}

MT=(A4−B4)T=(AT)4−(BT)4{M^T} = {({A^4} - {B^4})^T} = {({A^T})^4} - {({B^T})^4}

=A4−(−B)4=A4−B4=M= {A^4} - {( - B)^4} = {A^4} - {B^4} = M

(B) M=AB−BAM = AB - BA

MT=(AB−BA)T=(AB)T−(BA)T{M^T} = {(AB - BA)^T} = {(AB)^T} - {(BA)^T}

=BTAT−ATBT= {B^T}{A^T} - {A^T}{B^T}

=−BA−A(−B)= - BA - A( - B)

=AB−BA=M= AB - BA = M

(C) M=B5−A5M = {B^5} - {A^5}

MT=(BT)5−(AT)5=−(B5+A5)≠−M{M^T} = {({B^T})^5} - {({A^T})^5} = - ({B^5} + {A^5}) \ne - M

(D) M=AB+BAM = AB + BA

MT=(AB)T+(BA)T{M^T} = {(AB)^T} + {(BA)^T}

=BTAT+ATBT=−BA−AB=−M= {B^T}{A^T} + {A^T}{B^T} = - BA - AB = - M

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