First Principles AI

Question 4

If $A=\begin{bmatrix}1&2&-3\\0&-5&0\\-2&-2&7\end{bmatrix}$ and $B=\begin{bmatrix}-5&-2&5\\-3&-1&4\\-2&-1&2\end{bmatrix}$ , then find:

$A_{31},\,A_{32},\,A_{33}$ and $|A|$

$B_{31},\,B_{32},\,B_{33}$ and $|B|$

Question 5

Find the values of $x$ if:

$\left|\begin{matrix}2&1&x\\-1&-4&-3\\x&1&0\end{matrix}\right|=5$

$\left|\begin{matrix}1&x-1&3\\-1&x+1&2\\2&-3&x\end{matrix}\right|=9$

$\left|\begin{matrix}1&1&1\\2&x&2\\3&6&x\end{matrix}\right|=0$

Question 6

Find $|AA^T|$ and $|A^TA|$ if:

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

$A=\begin{bmatrix}3&1\\2&2\\1&3\end{bmatrix}$

Question 7

If $A$ is a square matrix of order 3, then show that $|kA|=k^3|A|$ .

Question 8

Find the values of $\lambda$ if $A$ and $B$ are singular.

$A=\begin{bmatrix}4&2&3\\7&\lambda&6\\2&3&1\end{bmatrix}$

$B=\begin{bmatrix}-2&4&5\\1&-2&1\\2&\lambda&0\end{bmatrix}$

Question 9

Find the inverse of $A=\begin{bmatrix}1&2&1\\-5&0&4\\5&4&0\end{bmatrix}$ and show that $A^{-1}A=I_3$ .

Question 10

Verify that $(AB)^T=B^TA^T$ if:

$A=\begin{bmatrix}1&-1&2\\0&-3&1\end{bmatrix}$ and $B=\begin{bmatrix}1&1\\-3&-2\\0&1\end{bmatrix}$

$A=\begin{bmatrix}1&2\\1&4\\2&1\end{bmatrix}$ and $B=\begin{bmatrix}1&-3\\-2&1\end{bmatrix}$

Questions