If A is nonsingular matrix then AB = AC implies B = C true or false

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If A is nonsingular matrix then AB = AC implies B = C true or false

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Arya 10 months 2021-07-19T03:36:44+00:00 1 Answers 0 views 0

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    2021-07-19T03:38:02+00:00

    \textbf{Given:}

    \textsf{A is non-singular matrix}

    \textbf{To find:}

    \textsf{The following is true or false}

    \mathsf{AB=AC\;\implies\;B=C}

    \textbf{Solution:}

    \mathsf{Since\;A\;is\;non-singular,\;A^{-1}\;exists}

    \mathsf{Consider,}

    \mathsf{AB=AC}

    \mathsf{Multiply\;bothsides\;by\;A^{-1}}

    \mathsf{A^{-1}(AB)=A^{-1}(AC)}

    \textsf{We know that,}

    \textbf{Matrix multiplication is associative}

    \implies\mathsf{(A^{-1}A)B=(A^{-1}A)C}

    \mathsf{But\;AA^{-1}=A^{-1}A=I,\,identity\;matrix}

    \implies\mathsf{(I)B=(I)C}

    \implies\boxed{\mathsf{B=C}}

    \therefore\mathsf{AB=AC\;\implies\;B=C\;is\;true}

    \textbf{Find more:}

    If A, B be real 2×2 matrices such that AA^T+BB^T =0,can A and B be invertible?

    https://brainly.in/question/19653584

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