Answer
The expression is defined, and its value is:
$\hphantom {2A^T + C=.} | 1 \space \space \space \space 8| $
$2A^T + C =| 5 \space \space 11| $
$ \hphantom {2A^T + C=.} | 6 \space \space 13| $
Work Step by Step
1. Check if the expression is defined.
For a sum operation, both matrices must have the same dimensions.
Since $A$ is a $2 \times 3$ matrix, $A^T$ is a $3 \times 2$ one.
$C$ is a $3 \times 2$ matrix.
Therefore, $2A^T + C$ is a defined expression.
2. Evaluate it:
- Transpose A : ($A_{ij} = A^T_{ji}$)
$A = | 1 \space \space 2 \space \space 3|$
$ \hphantom {.....} | 4 \space \space 5 \space \space 6|$
$ \hphantom {A^T =.} | 1 \space \space 4|$
$A^T = | 2 \space \space 5|$
$ \hphantom {A^T =.} | 3 \space \space 6|$
- Evaluate ($2A^T$)
$\hphantom {22A^T =.} | 1 \space \space 4|$
$2A^T =2| 2 \space \space 5|$
$ \hphantom {22A^T =.} | 3 \space \space 6|$
$\hphantom {2A^T =.} | 1 * 2 \space \space 4*2|$
$2A^T =| 2*2 \space \space 5*2|$
$ \hphantom {2A^T =.} | 3*2 \space \space 6*2|$
$\hphantom {2A^T =.} | 2 \space \space \space \space 8|$
$2A^T =| 4 \space \space 10|$
$ \hphantom {2A^T =.} | 6 \space \space 12|$
- Calculate $(2A^T + C)$
$\hphantom {2A^T + C=.} | 2 \space \space \space \space 8| \hphantom {...}|-1 \space \space 0|$
$2A^T + C =| 4 \space \space 10| + |1 \hphantom {....}1|$
$ \hphantom {2A^T + C=.} | 6 \space \space 12| \hphantom {...}|0 \hphantom {....} 1|$
$\hphantom {2A^T + C=.} | 2 - 1 \space \space \space \space 8 + 0| $
$2A^T + C =| 4 + 1 \space \space 10 + 1| $
$ \hphantom {2A^T + C=.} | 6 + 0 \space \space 12 + 1| $
$\hphantom {2A^T + C=.} | 1 \space \space \space \space 8| $
$2A^T + C =| 5 \space \space 11| $
$ \hphantom {2A^T + C=.} | 6 \space \space 13| $
