Trent Carpenter

2021-01-31

Let A,B and C be square matrices such that AB=AC , If $A\ne 0$ , then B=C.

Is this True or False?Explain the reasosing behind the answer.

Is this True or False?Explain the reasosing behind the answer.

unett

Skilled2021-02-01Added 119 answers

Step 1

Let A, B, and C be square matrices such that AB=AC.

If$A\ne 0$ , then B=C.

Determine true or false.

Step 2

Let A, B, and C be square matrices such that AB=AC.

If$A\ne 0$ , 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,

${A}^{-1}AB={A}^{-1}AC$

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

$I\times B=I\times C$

B=C

If A is not invertible then this statement is false.

Counter example:

$A=\left[\begin{array}{cc}2& -3\\ -4& 6\end{array}\right],B=\left[\begin{array}{cc}8& 4\\ 5& 5\end{array}\right],C=\left[\begin{array}{cc}5& -2\\ 3& 1\end{array}\right]$

Here,$det|A|=2\times 6-(-4)\times (-3)$

=12-12

=0

That is A is not invertible.

Now, find AB and AC,

$AB=\left[\begin{array}{cc}2& -3\\ -4& 6\end{array}\right]\left[\begin{array}{cc}8& 4\\ 5& 5\end{array}\right]$

$=\left[\begin{array}{cc}16-15& 8-15\\ -32+30& -16+30\end{array}\right]$

$=\left[\begin{array}{cc}1& -7\\ -2& 14\end{array}\right]$

$AC=\left[\begin{array}{cc}2& -3\\ -4& 6\end{array}\right]\left[\begin{array}{cc}5& -2\\ 3& 1\end{array}\right]$

$=\left[\begin{array}{cc}10-9& -4-3\\ -20+18& 8+6\end{array}\right]$

$=\left[\begin{array}{cc}1& -7\\ -2& 14\end{array}\right]$

That is, AB=AC but B is not the same as C.

Let A, B, and C be square matrices such that AB=AC.

If

Determine true or false.

Step 2

Let A, B, and C be square matrices such that AB=AC.

If

This is true only if matrix A is invertible.

If A is invertible then

Multiply given equationAB=AC by

B=C

If A is not invertible then this statement is false.

Counter example:

Here,

=12-12

=0

That is A is not invertible.

Now, find AB and AC,

That is, AB=AC but B is not the same as C.

Jeffrey Jordon

Expert2022-01-30Added 2605 answers

Answer is given below (on video)

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