Answer
The expression: "$AC + B$" is defined, and the resultin matrix is:
$AC + B =| 2 \space \space \space \space 4| $
$ \hphantom {2A^T + c=.} | 1 \space \space 12| $
Work Step by Step
1. Check if the expression is defined
For a matrix multiplication ($AC$), the length of the row in the first matrix must be equal to the length of the column in the second one.
$A$ row length: $3$
$C$ column length: $3$
Therefore, the expression is defined and the multiplication is possible.
$A$ is a $2 \times 3$ matrix, and $C$ is a $3 \times 2$ one.
So, the resulting matrix is going to be a $2 \times 2$
For a matrix addition ($AC$ + $B$), both matrices must have the same dimensions:
$AC$ is $2 \times 2$ and $B$ is $2 \times 2$.
- The expression is defined and can be evaluated:
2. Evaluate it.
First, we are going to calculate the entries for the $AC$ matrix:
$(AC)_{11} = (A_{11} \times C_{11} + A_{12} \times C_{21} + A_{13} \times C_{31})$
$(AC)_{12} = (A_{11} \times C_{12} + A_{12} \times C_{22} + A_{13} \times C_{32})$
$(AC)_{21} = (A_{21} \times C_{11} + A_{22} \times C_{21} + A_{23} \times C_{31})$
$(AC)_{22} = (A_{21} \times C_{12} + A_{22} \times C_{22} + A_{23} \times C_{32})$
$(AC)_{11} = 1 \times (-1) + 2 \times 1 + 3 \times 0 = 1$
$(AC)_{12} = 1 \times 0 + 2 \times 1 + 3 \times 1 = 5$
$(AC)_{21} = 4 \times (-1) + 5 \times 1 + 6 \times 0 = 1$
$(AC)_{22} = 4 \times 0 + 5 \times 1 + 6 \times 1 = 11$
Finally, we can add the matrices:
$(AC + B)_{11} = AC_{11} + B_{11} = 1 + 1 = 2$
$(AC + B)_{12} = AC_{12} + B_{12} = 5 + (-1) = 4$
$(AC + B)_{21} = AC_{21} + B_{21} = 1 + 0 = 1$
$(AC + B)_{22} = AC_{22} + B_{22} = 11 + 1 = 12$
$AC + B =| 2 \space \space \space \space 4| $
$ \hphantom {2A^T + c=.} | 1 \space \space 12| $
