Question

(b) Give an example of nonzero 2×2 matrices A, B, C such that AB=AC, but B C.

Answer 1

Example of matrices such hat
`AB = AC, B\\eq C`

Explanation:

We have two give example of 2×2 matrices such that AB = AC BUT B ≠ C.

Example:

`A =\left[\begin{array}{ccc}1&0\\0&0\end{array}\right]\\\\B = \left[\begin{array}{ccc}1&1\\1&2\end{array}\right], C = \left[\begin{array}{ccc}1&1\\1&3\end{array}\right]`

Solving:

`AB = \left[\begin{array}{ccc}1+0&1+0\\0+0&0+0\end{array}\right] = \left[\begin{array}{ccc}1&1\\0&0\end{array}\right]`

`AC = \left[\begin{array}{ccc}1+0&1+0\\0+0&0+0\end{array}\right] = \left[\begin{array}{ccc}1&1\\0&0\end{array}\right]`

Hence,
`AB = AC, B\\eq C`