Answer
The expression is defined and its value is:
$\hphantom {A^TB=.} | 1 \space \space 3| $
$A^TB =| 2 \space \space 3| $
$ \hphantom {A^TB=.} |3 \space \space 3| $
Work Step by Step
1. Check if the expression is defined:
$A$ is a $2 \times 3$ matrix, so $A^T$ is a $3 \times 2$ one.
For a matrix multiplication, the length of the row in $A^T$ must be equal to the length of the column in B.
Length of the row in A: $2$
Length of the column in B: $2$
Therefore, this expression in undefined.
2. Evaluate $A^TB$
$(A^TB)_{11} = (A^T_{11} \times B_{11} + A^T_{12} \times B_{21})$
$(A^TB)_{12} = (A^T_{11} \times B_{12} + A^T_{12} \times B_{22})$
$(A^TB)_{21} = (A^T_{21} \times B_{11} + A^T_{22} \times B_{21})$
$(A^TB)_{22} = (A^T_{21} \times B_{12} + A^T_{22} \times B_{22})$
$(A^TB)_{31} = (A^T_{31} \times B_{11} + A^T_{32} \times B_{21})$
$(A^TB)_{32} = (A^T_{31} \times B_{12} + A^T_{32} \times B_{22})$
Since: $A^T_{ij} = A_{ji}$:
$(A^TB)_{11} = (A_{11} \times B_{11} + A_{21} \times B_{21})$
$(A^TB)_{12} = (A_{11} \times B_{12} + A_{21} \times B_{22})$
$(A^TB)_{21} = (A_{12} \times B_{11} + A_{22} \times B_{21})$
$(A^TB)_{22} = (A_{12} \times B_{12} + A_{22} \times B_{22})$
$(A^TB)_{31} = (A_{13} \times B_{11} + A_{23} \times B_{21})$
$(A^TB)_{32} = (A_{13} \times B_{12} + A_{23} \times B_{22})$
$(A^TB)_{11} =1 * 1 + 4 * 0 = 1$
$(A^TB)_{12} = 1 * (-1) + 4 * 1 = 3$
$(A^TB)_{21} =2 * 1 + 5 * 0 = 2$
$(A^TB)_{22} = 2 * (-1) + 5 * 1 = 3$
$(A^TB)_{31} = 3 * 1 + 6 * 0 = 3$
$(A^TB)_{32} = 3 * (-1) + 6 * 1 = 3$
$\hphantom {A^TB=.} | 1 \space \space 3| $
$A^TB =| 2 \space \space 3| $
$ \hphantom {A^TB=.} |3 \space \space 3| $
