Let A,B and C be square matrices such that AB=AC , If A≠0 , then B=C. Is this True or False?Explain the reasosing behind the answer.

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Let A,B and C be square matrices such that AB=AC , If A≠0 , then B=C.  Is this True or False?Explain the reasosing behind the answer.

unett · Accepted Answer

Step 1  Let A, B, and C be square matrices such that AB=AC.  If A≠0, then B=C.  Determine true or false.  Step 2  Let A, B, and C be square matrices such that AB=AC.  If A≠0 , then B=C.  This is true only if matrix A is invertible.  If A is invertible then A−1 exist.  Multiply given equationAB=AC by A−1 on both sides,  A−1AB=A−1AC  (A−1A)B=(A−1A)C  I×B=I×C  B=C  If A is not invertible then this statement is false.  Counter example:  A=[2−3−46],B=[8455],C=[5−231]  Here, det|A|=2×6−(−4)×(−3)  =12-12  =0  That is A is not invertible.  Now, find AB and AC,  AB=[2−3−46][8455]  =[16−158−15−32+30−16+30]  =[1−7−214]  AC=[2−3−46][5−231]  =[10−9−4−3−20+188+6]  =[1−7−214]  That is, AB=AC but B is not the same as C.

Jeffrey Jordon · Accepted Answer

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Let A,B and C be square matrices such that AB=AC , If A neq 0 , then B=C. Is this True or False?Explain the reasosing behind the answer.

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