Question

If A is nonsingular matrix then AB = AC implies B = C true or false
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Answer 1

$\textbf{Given:}$

$\textsf{A is non-singular matrix}$

$\textbf{To find:}$

$\textsf{The following is true or false}$

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

$\textbf{Solution:}$

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

$\mathsf{Consider,}$

$\mathsf{AB=AC}$

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

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

$\textsf{We know that,}$

$\textbf{Matrix multiplication is associative}$

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

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

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

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

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

$\textbf{Find more:}$

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

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