Can anyone explain me the product of a matrix by its transpose?

Question

Please solve it and explain it to me

$$ A=\begin{bmatrix} 1 & 0\\ -1 & 1\\ \end{bmatrix},\quad AA^T = \ ? $$

Answer 1

$$A=\begin{bmatrix} 1 & 0\\ -1 & 1\\ \end{bmatrix}, \qquad A^T=\begin{bmatrix} 1 & -1\\ 0 & 1\\ \end{bmatrix}$$

Then $AA^T$ can be calculated by simply multiplying each row by each column

Answer 2

You have $A=\begin{bmatrix} 1 & 0\\ -1 & 1\\ \end{bmatrix}$

$A^T = \begin{bmatrix}1 &&-1\\0&&1\end{bmatrix}$

$A\cdot A^T=\begin{bmatrix} 1 & 0\\ -1 & 1\\ \end{bmatrix}\cdot \begin{bmatrix}1 &&-1\\0&&1\end{bmatrix}$

$ = \begin{bmatrix}1&&-1\\-1&&2\end{bmatrix}$

Answer 3

Try using the following: $$ \begin{bmatrix} a_{11} &a_{12} \\ a_{21} &a_{22} \end{bmatrix} \begin{bmatrix} b_{11} &b_{12} \\ b_{21} &b_{22} \end{bmatrix} = \begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} &a_{11}b_{12} + a_{12}b_{22} \\ a_{21}b_{11} + a_{22}b_{21} &a_{21}b_{12} + a_{22}b_{22} \end{bmatrix} $$