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.

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.

Question
Matrices
asked 2021-01-31
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.

Answers (1)

2021-02-01
Step 1
Let A, B, and C be square matrices such that AB=AC.
If \(A \neq 0\), then B=C.
Determine true or false.
Step 2
Let A, B, and C be square matrices such that AB=AC.
If \(A \neq 0\) , 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:
\(A=\begin{bmatrix}2 & -3 \\-4 & 6 \end{bmatrix} , B=\begin{bmatrix}8 & 4 \\5 & 5 \end{bmatrix} , C=\begin{bmatrix}5 & -2 \\3 & 1 \end{bmatrix}\)
Here, \(det|A|=2 \times 6 -(-4) \times (-3)\)
=12-12
=0
That is A is not invertible.
Now, find AB and AC,
\(AB=\begin{bmatrix}2 & -3 \\-4 & 6 \end{bmatrix}\begin{bmatrix}8 & 4 \\5 & 5 \end{bmatrix}\)
\(=\begin{bmatrix}16-15 & 8-15 \\-32+30 & -16+30 \end{bmatrix}\)
\(=\begin{bmatrix}1 & -7 \\-2 & 14 \end{bmatrix}\)
\(AC=\begin{bmatrix}2 & -3 \\-4 & 6 \end{bmatrix}\begin{bmatrix}5 & -2 \\3 & 1 \end{bmatrix}\)
\(=\begin{bmatrix}10-9 & -4-3 \\-20+18 & 8+6 \end{bmatrix}\)
\(=\begin{bmatrix}1 & -7 \\-2 & 14 \end{bmatrix}\)
That is, AB=AC but B is not the same as C.
0

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