If A is nonsingular matrix then AB = AC implies B = C true or false
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1 thought on “If A is nonsingular matrix then AB = AC implies B = C true or false<br />​”

1. [tex]\textbf{Given:}[/tex]

   [tex]\textsf{A is non-singular matrix}[/tex]

   [tex]\textbf{To find:}[/tex]

   [tex]\textsf{The following is true or false}[/tex]

   [tex]\mathsf{AB=AC\;\implies\;B=C}[/tex]

   [tex]\textbf{Solution:}[/tex]

   [tex]\mathsf{Since\;A\;is\;non-singular,\;A^{-1}\;exists}[/tex]

   [tex]\mathsf{Consider,}[/tex]

   [tex]\mathsf{AB=AC}[/tex]

   [tex]\mathsf{Multiply\;bothsides\;by\;A^{-1}}[/tex]

   [tex]\mathsf{A^{-1}(AB)=A^{-1}(AC)}[/tex]

   [tex]\textsf{We know that,}[/tex]

   [tex]\textbf{Matrix multiplication is associative}[/tex]

   [tex]\implies\mathsf{(A^{-1}A)B=(A^{-1}A)C}[/tex]

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

   [tex]\implies\mathsf{(I)B=(I)C}[/tex]

   [tex]\implies\boxed{\mathsf{B=C}}[/tex]

   [tex]\therefore\mathsf{AB=AC\;\implies\;B=C\;is\;true}[/tex]

   [tex]\textbf{Find more:}[/tex]

   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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