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Provide solution for rd sharma maths class 12 chapter adjoint and inverse of a matrix exercise multiple choice question question 14

Answers (1)

option (b) A^{2}+B^{2}

Given:  A & B are square matrices and B = - A^{-1} BA

Hint: to solve the question you must know the formula of \left (A+B \right )^{2}

Solution:

 B = - A^{-1} BA

Multiply both sides with A

 \begin{aligned} &A B=A\left(-A^{-1} B A\right) \\ &A B=-\left(A A^{-1}\right) B A \\ &A B=-B A \quad\left(\therefore A A^{-1}=I\right) \\ &\text { Now, }(A+B)^{2}=(A+B)(A+B) \\ &=A^{2}+A B+B A+B^{2} \\ &=A^{2}-B A+B A+B^{2} \quad(\text { since } A B=-\text { BA }) \\ &(A+B)^{2}=A^{2}+B^{2} \end{aligned}

Hence, option b is correct

 

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