First Principles AI

Solution

Given:

A=\begin{bmatrix}-1&2&3\\1&0&2\\-3&5&3\end{bmatrix}

First find the transpose:

A^T=\begin{bmatrix}-1&1&-3\\2&0&5\\3&2&3\end{bmatrix}

1) $A+A^T$

A+A^T= \begin{bmatrix}-2&3&0\\3&0&7\\0&7&6\end{bmatrix}

2) $A-A^T$

A-A^T= \begin{bmatrix}0&1&6\\-1&0&-3\\-6&3&0\end{bmatrix}

3) $AA^T$

AA^T= \begin{bmatrix}14&5&22\\5&5&3\\22&3&43\end{bmatrix}

4) $A^TA$

A^TA= \begin{bmatrix}11&-17&-10\\-17&29&21\\-10&21&22\end{bmatrix}

5) $(A^T)^T$

Transpose of the transpose returns the original matrix:

(A^T)^T=A= \begin{bmatrix}-1&2&3\\1&0&2\\-3&5&3\end{bmatrix}

\boxed{A+A^T=\begin{bmatrix}-2&3&0\\3&0&7\\0&7&6\end{bmatrix}}

\boxed{A-A^T=\begin{bmatrix}0&1&6\\-1&0&-3\\-6&3&0\end{bmatrix}}

\boxed{AA^T=\begin{bmatrix}14&5&22\\5&5&3\\22&3&43\end{bmatrix}}

\boxed{A^TA=\begin{bmatrix}11&-17&-10\\-17&29&21\\-10&21&22\end{bmatrix}}

\boxed{(A^T)^T=A}

Solution