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.

Trent Carpenter 2021-01-31 Answered
Let A,B and C be square matrices such that AB=AC , If A0 , then B=C.
Is this True or False?Explain the reasosing behind the answer.
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unett
Answered 2021-02-01 Author has 119 answers
Step 1
Let A, B, and C be square matrices such that AB=AC.
If A0, then B=C.
Determine true or false.
Step 2
Let A, B, and C be square matrices such that AB=AC.
If A0 , then B=C.
This is true only if matrix A is invertible.
If A is invertible then A1 exist.
Multiply given equationAB=AC by A1 on both sides,
A1AB=A1AC
(A1A)B=(A1A)C
I×B=I×C
B=C
If A is not invertible then this statement is false.
Counter example:
A=[2346],B=[8455],C=[5231]
Here, det|A|=2×6(4)×(3)
=12-12
=0
That is A is not invertible.
Now, find AB and AC,
AB=[2346][8455]
=[161581532+3016+30]
=[17214]
AC=[2346][5231]
=[1094320+188+6]
=[17214]
That is, AB=AC but B is not the same as C.
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Jeffrey Jordon
Answered 2022-01-30 Author has 2064 answers

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