Lista de Exercícios de Matrizes 01) Construa a matriz real q...

Lista de Exercícios de Matrizes 01) Construa a matriz real quadrada A de ordem 3, definida por: $\quad a_{i j}=\left\{\begin{array}{l}2^{\mathrm{i}+\mathrm{j}} \text { se } i<j \\ i^{2}-j+1 \text { se } i \geq j\end{array}\right.$ 02) Sendo $M=\left(\begin{array}{ccc}1 & 2 & 3 \\ -1 & 0 & -2 \\ 4 & -3 & 5\end{array}\right), N=\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)$ e $P=\left(\begin{array}{ccc}0 & -1 & 1 \\ -2 & 0 & 1 \\ -3 & 2 & 0\end{array}\right)$, calcule: a) $N-P+M$ b) $2 M-3 N-P$ c) $\quad \mathrm{N}-2(\mathrm{M}-\mathrm{P})$ 03) Calcule a matriz $X$, sabendo que $A=\left(\begin{array}{cc}1 & 2 \\ -1 & 0 \\ 4 & 3\end{array}\right), B=\left(\begin{array}{ccc}5 & 1 & 3 \\ -2 & 0 & 2\end{array}\right)$ e $(X+A)^{t}=B$. 04) Dadas as matrizes $A=\left[\begin{array}{ll}a & 0 \\ 0 & a\end{array}\right]$ e $B=\left[\begin{array}{ll}1 & b \\ b & 1\end{array}\right]$, determine a e $b$, de modo que $A B=I$, em que $\mathrm{I}$ é a matriz identidade. 05) Dadas as matrizes $A=\left[\begin{array}{cc}1 & -2 \\ 0 & 3\end{array}\right]$ e $B=\left[\begin{array}{cc}1 & -3 \\ 2 & 0\end{array}\right]$. Calcule: a) $\mathrm{A}^{2}$ b) $A^{3}$ c) $A^{2} B$ d) $A^{2}+3 B$ 06) Dadas as matrizes $A=\left(\begin{array}{cc}-1 & 2 \\ 3 & 1\end{array}\right)$ e $B=\left(\begin{array}{cc}2 & -1 \\ 4 & 3\end{array}\right)$, calcule $A B+B^{t}$ 07) Resolva a equação:

$$\left(\begin{array}{cc} 2 x & -3 \\ x-1 & y \end{array}\right) \cdot\left(\begin{array}{cc} 2 & x \\ -1 & y \end{array}\right)=\left(\begin{array}{cc} 11 & 2 x^{2}-3 y \\ 2 x-y-2 & 11 \end{array}\right)$$