Question

Find AC when

$A\:=\:\begin{gathered}\left[\begin{array}{ccc} 2&1 \\ 0& - 2\end{array}\right]\end{gathered}$
$C\:=\:\begin{gathered}\left[\begin{array}{ccc} - 3&2\\ - 1&4\end{array}\right]\end{gathered}$

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Answer 1

Multiply - Matrix

The matrix multiplication can be done only if the number of columns of the first matrix is equal to the number of rows of the second matrix.

Given matrices are,

$\longrightarrow A = \left[\begin{array}{cc}2 & 1 \\ 0 & - 2\end{array}\right]$

$\longrightarrow C = \left[\begin{array}{cc} -3 & 2\\ - 1 & 4\end{array}\right]$

We need to find out the value of AC, it means we have to multiply both the matrices.

Solution:

Multiply each row in the first matrix by each column in the second matrix, then simplify each element of the matrix by multiplying out all the expressions.

$AC = A \times C$

$\implies \left[\begin{array}{cc}2 & 1 \\ 0 & - 2\end{array}\right] \left[\begin{array}{cc} -3 & 2\\ - 1 & 4\end{array}\right]$

$\implies \left[\begin{array}{cc} 2(3) + 1(-1) & 2(2) + 1(4) \\ 0(-3) - 2(-1) & 0(2) - 2(4)\end{array}\right]$

$\implies \left[\begin{array}{cc} - 6 -1 &4 + 4\\ 0 + 2& 0 - 8\end{array}\right]$

$\implies \left[\begin{array}{cc} -7 & 8 \\ 2 & -8\end{array}\right]$

Therefore the required answer is:

$\boxed{AC = \left[\begin{array}{cc} -7 & 8 \\ 2 & -8\end{array}\right]}$

Answer 2

Therefore the required answer is:

$\begin{gathered}\boxed{AC = \left[\begin{array}{cc} -7 & 8 \\ 2 & -8\end{array}\right]}\end{gathered}AC=[−728−8]$