91Ó°ÊÓ

In Exercises \(9-16,\) find: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{ll} 1 & 3 \\ 3 & 4 \\ 5 & 6 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & -1 \\ 3 & -2 \\ 0 & 1 \end{array}\right] $$

Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$\begin{aligned}2 w+x-y &=3 \\\w-3 x+2 y &=-4 \\\3 w+x-3 y+z &=1 \\\w+2 x-4 y-z &=-2\end{aligned}$$

In Exercises \(9-12,\) write the system of linear equations represented by the augmented matrix. Use \(x, y, z,\) and, if necessary, \(w, x, y,\) and \(z,\) for the variables. \(\left[\begin{array}{rrrr|r}1 & 1 & 4 & 1 & 3 \\ -1 & 1 & -1 & 0 & 7 \\ 2 & 0 & 0 & 5 & 11 \\ 0 & 0 & 12 & 4 & 5\end{array}\right]\)

Use Cramer's rule to solve each system or to determine that the system is inconsistent or contains dependent equations. $$ \begin{aligned}&x+y=7\\\&x-y=3\end{aligned} $$

find the products \(A B\) and \(B A\) to determine whether \(B\) is the multiplicative inverse of \(A\). $$ A=\left[\begin{array}{rrrr} 0 & 0 & -2 & 1 \\ -1 & 0 & 1 & 1 \\ 0 & 1 & -1 & 0 \\ 1 & 0 & 0 & -1 \end{array}\right], \quad B=\left[\begin{array}{llll} 1 & 2 & 0 & 3 \\ 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 1 & 2 & 0 & 2 \end{array}\right] $$

find the products \(A B\) and \(B A\) to determine whether \(B\) is the multiplicative inverse of \(A\). $$ A=\left[\begin{array}{rrrr} 1 & -2 & 1 & 0 \\ 0 & 1 & -2 & 1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 1 \end{array}\right], \quad B=\left[\begin{array}{llll} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 1 \end{array}\right] $$

In Exercises \(9-12,\) write the system of linear equations represented by the augmented matrix. Use \(x, y, z,\) and, if necessary, \(w, x, y,\) and \(z,\) for the variables. \(\left[\begin{array}{rrrr|r}4 & 1 & 5 & 1 & 6 \\ 1 & -1 & 0 & -1 & 8 \\ 3 & 0 & 0 & 7 & 4 \\ 0 & 0 & 11 & 5 & 3\end{array}\right]\)

In Exercises \(9-16,\) find: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{rrr} 3 & 1 & 1 \\ -1 & 2 & 5 \end{array}\right], \quad B=\left[\begin{array}{rrr} 2 & -3 & 6 \\ -3 & 1 & -4 \end{array}\right] $$

Use Gaussian elimination to find the complete solution to each system of equations, or show that none exists. $$\begin{aligned}&2 w-x+3 y+z=0\\\&3 w+2 x+4 y-z=0\\\&5 w-2 x-2 y-z=0\\\&2 w+3 x-7 y-5 z=0\end{aligned}$$

Use Cramer's rule to solve each system or to determine that the system is inconsistent or contains dependent equations. $$ \begin{array}{r}2 x+y=3 \\\x-y=3\end{array} $$