Matrix-Multiplication

Question

I have to matrices:

$$A=\pmatrix{1&a&1\\1&0&a\\1&2&0} ; \quad B= \pmatrix{1&b&3\\2&1&0}$$

The task is to determine $AB, AB^T, BA$

I think i cannot calculate the matrix of $AB$ because $\text{Columns} \ A = 3$ is not $\text{Rows} \ B = 2$

But i can calculate $BA$:

$$BA=\pmatrix{4+b&6+a&1+ba\\3&2a&2+a}$$

Now my question is, what is meant with $ AB^T$ ? Thanks

Answer 1

$A\times B^T$ means the matrix $A$ multiplied by the transpose of $B$. Given some matrix $A$, the transpose, $A^T$, is a matrix such that the columns of $A$ are the rows of $A^T$ and the rows of $A$ are the columns of $A^T$. Thus we see that $$B^T= \left(\begin{matrix} 1 & 2 \\ b & 1 \\ 3 & 0 \\ \end{matrix}\right) .$$ You can now evaluate $A\times B^T.$

Answer 2

Hint: $B^{T}$ means transpose of $B$. You can read about transpose here.