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BCECE Engineering BCECE Engineering Solved Paper-2015

  • question_answer
    If A is a square matrix such that \[{{\text{A}}^{\text{2}}}\text{ }=\text{ A}\] and  \[\text{B }=\text{ I }-\text{ A,}\], then \[AB+BA+I-{{(I-A)}^{2}}\] is equal to

    A) A                                              

    B) 2A    

    C) -A                                          

    D) \[l-A\]

    Correct Answer: A

    Solution :

    We have, \[{{A}^{2}}=A\]and \[B=I-A\] \[\therefore \]  \[AB+BA+I-{{(I-A)}^{2}}\] \[=A(I-A)+(I-A)A+I-(I-A)(I-A)\] \[=A-{{A}^{2}}+A-{{A}^{2}}+I-(I-2A+{{A}^{2}})\] \[=A-A+A-A+I-(I-2A+A)\] \[[\because {{A}^{2}}=A]\] \[=A\]


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