91Ó°ÊÓ

Find the following matrices: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{rrr} {2} & {-10} & {-2} \\ {14} & {12} & {10} \\ {4} & {-2} & {2} \end{array}\right], \quad B=\left[\begin{array}{rrr} {6} & {10} & {-2} \\ {0} & {-12} & {-4} \\ {-5} & {2} & {-2} \end{array}\right] $$

Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$ \left\\{\begin{array}{l} {2 x+y-z=2} \\ {3 x+3 y-2 z=3} \end{array}\right. $$

Perform each matrix row operation and write the new matrix. $$ \left[\begin{array}{rrr|r} {1} & {-3} & {2} & {0} \\ {3} & {1} & {-1} & {7} \\ {2} & {-2} & {1} & {3} \end{array}\right] \quad-3 R_{1}+R_{2} $$

use the fact that if \(\boldsymbol{A}=\left[\begin{array}{ll}{\boldsymbol{a}} & {\boldsymbol{b}} \\ {\boldsymbol{c}} & {\boldsymbol{d}}\end{array}\right],\) then \(A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{cc}{d} & {-b} \\ {-c} & {a}\end{array}\right]\) to find the inverse of each matrix, if possible. Check that \(A A^{-1}=I_{2}\) and \(A^{-1} A=I_{2}\) $$ A=\left[\begin{array}{rr} {3} & {-1} \\ {-4} & {2} \end{array}\right] $$

Use Cramer’s Rule to solve each system. $$\left\\{\begin{array}{l}{4 x-5 y=17} \\\\{2 x+3 y=3}\end{array}\right.$$

Use Cramer’s Rule to solve each system. $$\left\\{\begin{array}{l}{3 x+2 y=2} \\\\{2 x+2 y=3}\end{array}\right.$$

Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$ \left\\{\begin{array}{r} {3 x+2 y-z=5} \\ {x+2 y-z=1} \end{array}\right. $$

Perform each matrix row operation and write the new matrix. $$ \left[\begin{array}{rrr|r} {1} & {-1} & {5} & {-6} \\ {3} & {3} & {-1} & {10} \\ {1} & {3} & {2} & {5} \end{array}\right] \quad-3 R_{1}+R_{2} $$

Find the following matrices: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{rrr} {6} & {-3} & {5} \\ {6} & {0} & {-2} \\ {-4} & {2} & {-1} \end{array}\right], \quad B=\left[\begin{array}{rrr} {-3} & {5} & {1} \\ {-1} & {2} & {-6} \\ {2} & {0} & {4} \end{array}\right] $$

use the fact that if \(\boldsymbol{A}=\left[\begin{array}{ll}{\boldsymbol{a}} & {\boldsymbol{b}} \\ {\boldsymbol{c}} & {\boldsymbol{d}}\end{array}\right],\) then \(A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{cc}{d} & {-b} \\ {-c} & {a}\end{array}\right]\) to find the inverse of each matrix, if possible. Check that \(A A^{-1}=I_{2}\) and \(A^{-1} A=I_{2}\) $$ A=\left[\begin{array}{ll} {2} & {-6} \\ {1} & {-2} \end{array}\right] $$