Chapter 6

Use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rrrr} 1 & -2 & -1 & -2 \\ 3 & -5 & -2 & -3 \\ 2 & -5 & -2 & -5 \\ -1 & 4 & 4 & 11 \end{array}\right] $$

Write the system of linear equations represented by the augmented matrix. (Use the variables \(x, y, z\), and \(w .)\) $$ \left[\begin{array}{rrrrr} 1 & 0 & 2 & -10 \\ 0 & 3 & -1 & 5 \\ 4 & 2 & 0 & 3 \end{array}\right] $$

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$ \left[\begin{array}{rrr} -3 & 0 & 0 \\ 7 & 11 & 0 \\ 1 & 2 & 2 \end{array}\right] $$

Use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rrrr} 4 & 8 & -7 & 14 \\ 2 & 5 & -4 & 6 \\ 0 & 2 & 1 & -7 \\ 3 & 6 & -5 & 10 \end{array}\right] $$

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$ \left[\begin{array}{rrr} 2 & -1 & 0 \\ 4 & 2 & 1 \\ 4 & 2 & 1 \end{array}\right] $$

Find (a) \(A B\), (b) \(B A\), and, if possible, (c) \(A^{2}\). (Note: \(A^{2}=A A\).) $$ A=\left[\begin{array}{ll} 1 & 2 \\ 4 & 2 \end{array}\right], B=\left[\begin{array}{rr} 2 & -1 \\ -1 & 8 \end{array}\right] $$

Write the augmented matrix for the system of linear equations. $$ \left\\{\begin{array}{l} 2 x-y=3 \\ 5 x+7 y=12 \end{array}\right. $$

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$ \left[\begin{array}{rrr} -2 & 2 & 3 \\ 1 & -1 & 0 \\ 0 & 1 & 4 \end{array}\right] $$

Find (a) \(A B\), (b) \(B A\), and, if possible, (c) \(A^{2}\). (Note: \(A^{2}=A A\).) $$ A=\left[\begin{array}{rr} 2 & -1 \\ 1 & 4 \end{array}\right], B=\left[\begin{array}{rr} 0 & 0 \\ 3 & -3 \end{array}\right] $$

Write the augmented matrix for the system of linear equations. $$ \left\\{\begin{array}{l} 8 x+3 y=25 \\ 3 x-9 y=12 \end{array}\right. $$

Find (a) \(A B\), (b) \(B A\), and, if possible, (c) \(A^{2}\). (Note: \(A^{2}=A A\).) $$ A=\left[\begin{array}{rrr} -1 & 2 & 3 \\ 4 & 1 & -1 \end{array}\right], B=\left[\begin{array}{rr} 1 & 3 \\ -1 & -2 \\ 2 & 4 \end{array}\right] $$

Write the augmented matrix for the system of linear equations. $$ \left\\{\begin{array}{r} x+10 y-3 z=2 \\ 5 x-3 y+4 z=0 \\ 2 x+4 y=6 \end{array}\right. $$

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$ \left[\begin{array}{rrr} 1 & 4 & -2 \\ 3 & 6 & -6 \\ -2 & 1 & 4 \end{array}\right] $$

Find (a) \(A B\), (b) \(B A\), and, if possible, (c) \(A^{2}\). (Note: \(A^{2}=A A\).) $$ A=\left[\begin{array}{rrr} 1 & -1 & 7 \\ 2 & -1 & 8 \\ 3 & 1 & -1 \end{array}\right], B=\left[\begin{array}{rrr} 1 & 1 & 2 \\ 2 & 1 & 1 \\ 1 & -3 & 2 \end{array}\right] $$

Write the augmented matrix for the system of linear equations. $$ \left\\{\begin{array}{rr} 2 x+3 y-z= & 8 \\ y+2 z= & -10 \\ x-2 y-3 z= & 21 \end{array}\right. $$

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$ \left[\begin{array}{rrr} -1 & 3 & 1 \\ 4 & 2 & 5 \\ -2 & 1 & 6 \end{array}\right] $$

Find (a) \(A B\), (b) \(B A\), and, if possible, (c) \(A^{2}\). (Note: \(A^{2}=A A\).) $$ A=\left[\begin{array}{lll} -4 & 2 & 3 \end{array}\right], B=\left[\begin{array}{l} 1 \\ 0 \\ 5 \end{array}\right] $$

Write the augmented matrix for the system of linear equations. Write the augmented matrix for the system of linear equations. $$ \left\\{\begin{array}{rr} 9 w-3 x+20 y+z= & 13 \\ 12 w-8 y= & 5 \\ w+2 x+3 y-4 z= & -2 \\ -w-x+y+z= & 1 \end{array}\right. $$

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$ \left[\begin{array}{rrr} 0.3 & 0.2 & 0.2 \\ 0.2 & 0.2 & 0.2 \\ -0.4 & 0.4 & 0.3 \end{array}\right] $$

Find (a) \(A B\), (b) \(B A\), and, if possible, (c) \(A^{2}\). (Note: \(A^{2}=A A\).) $$ A=\left[\begin{array}{llll} 3 & 2 & 1 & 0 \end{array}\right], B=\left[\begin{array}{l} 2 \\ 3 \\ 1 \\ 0 \end{array}\right] $$

Write the augmented matrix for the system of linear equations. $$ \left\\{\begin{array}{rr} w+2 x-3 y+z= & 18 \\ 3 w \quad-5 y=8 \\ w+x+y+2 z= & 15 \\ -w-x+2 y+z= & -3 \end{array}\right. $$

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$ \left[\begin{array}{rrr} 0.1 & 0.2 & 0.3 \\ -0.3 & 0.2 & 0.2 \\ 0.5 & 0.4 & 0.4 \end{array}\right] $$

(a) write the system of linear equations as a matrix equation \(A X=B\), and (b) use Gauss-Jordan elimination on the augmented matrix \([A: B]\) to solve for the matrix \(X\). $$ \left\\{\begin{array}{r} -x+y=4 \\ -2 x+y=0 \end{array}\right. $$

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$ \left[\begin{array}{rrr} 6 & 3 & -7 \\ 0 & 0 & 0 \\ 4 & -6 & 3 \end{array}\right] $$

(a) write the system of linear equations as a matrix equation \(A X=B\), and (b) use Gauss-Jordan elimination on the augmented matrix \([A: B]\) to solve for the matrix \(X\). $$ \left\\{\begin{array}{r} 2 x+3 y=5 \\ x+4 y=10 \end{array}\right. $$

Write the system of equations represented by the augmented matrix. Use backsubstitution to find the solution. (Use \(x, y, z\), and \(w .)\) $$ \left[\begin{array}{rrrrr} 1 & 2 & -1 & \vdots & 3 \\ 0 & 1 & -2 & \vdots & -3 \\ 0 & 0 & 1 & \vdots & 4 \end{array}\right] $$

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$ \left[\begin{array}{rrr} 5 & 0 & 3 \\ -4 & 0 & 8 \\ 3 & 0 & -6 \end{array}\right] $$

(a) write the system of linear equations as a matrix equation \(A X=B\), and (b) use Gauss-Jordan elimination on the augmented matrix \([A: B]\) to solve for the matrix \(X\). $$ \left\\{\begin{array}{r} x+2 y=3 \\ 3 x-y=2 \end{array}\right. $$

Write the system of equations represented by the augmented matrix. Use backsubstitution to find the solution. (Use \(x, y, z\), and \(w .)\) $$ \left[\begin{array}{rrrr} \mathbf{1} & \mathbf{3} & -\mathbf{1} & 15 \\ \mathbf{0} & \mathbf{1} & \mathbf{4} & -12 \\ 0 & 0 & 1 & -5 \end{array}\right] $$

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$ \left[\begin{array}{rrrr} 3 & 6 & -5 & 4 \\ -2 & 0 & 6 & 0 \\ 1 & 1 & 2 & 2 \\ 0 & 3 & -1 & -1 \end{array}\right] $$

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$ \left[\begin{array}{llll} 2 & 6 & 6 & 2 \\ 2 & 7 & 3 & 6 \\ 1 & 5 & 0 & 1 \\ 3 & 7 & 0 & 7 \end{array}\right] $$

(a) write the system of linear equations as a matrix equation \(A X=B\), and (b) use Gauss-Jordan elimination on the augmented matrix \([A: B]\) to solve for the matrix \(X\). $$ \left\\{\begin{aligned} 2 x-4 y+z &=0 \\ -x+3 y+z &=1 \\ x+y &=3 \end{aligned}\right. $$

Write the system of equations represented by the augmented matrix. Use backsubstitution to find the solution. (Use \(x, y, z\), and \(w .)\) $$ \left[\begin{array}{rrrrrr} 1 & 2 & -2 & 0 & \vdots & -1 \\ 0 & 1 & 1 & 2 & \vdots & 9 \\ 0 & 0 & 1 & 0 & \vdots & 2 \\ 0 & 0 & 0 & 1 & \vdots & -3 \end{array}\right] $$

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$ \left[\begin{array}{rrrr} 5 & 3 & 0 & 6 \\ 4 & 6 & 4 & 12 \\ 0 & 2 & -3 & 4 \\ 0 & 1 & -2 & 2 \end{array}\right] $$

(a) write the system of linear equations as a matrix equation \(A X=B\), and (b) use Gauss-Jordan elimination on the augmented matrix \([A: B]\) to solve for the matrix \(X\). $$ \left\\{\begin{aligned} x-2 y+3 z &=\quad 9 \\ -x+3 y-z &=-6 \\ 2 x-5 y+5 z &=17 \end{aligned}\right. $$

An augmented matrix that represents a system of linear equations (in variables \(x, y\), and \(z\) ) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. $$ \left[\begin{array}{rrrr} 1 & 0 & -4 \\ 0 & 1 & 6 \end{array}\right] $$

Use \(A^{-1}\) to decode the cryptogram. \(A=\left[\begin{array}{rrr}1 & -1 & 0 \\ 1 & 0 & -1 \\ -6 & 2 & 3\end{array}\right]\) \(9,-1,-9,38,-19,-19,28,-9,-19,-80,25,41\) \(-64,21,31,-7,-4,7\)

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$ \left[\begin{array}{rrrr} 1 & 4 & 3 & 2 \\ -5 & 6 & 2 & 1 \\ 0 & 0 & 0 & 0 \\ 3 & -2 & 1 & 5 \end{array}\right] $$

(a) write the system of linear equations as a matrix equation \(A X=B\), and (b) use Gauss-Jordan elimination on the augmented matrix \([A: B]\) to solve for the matrix \(X\). $$ \left\\{\begin{aligned} x+y-3 z &=-1 \\ -x+2 y &=1 \\ x-y+z &=2 \end{aligned}\right. $$

An augmented matrix that represents a system of linear equations (in variables \(x, y\), and \(z\) ) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. $$ \left[\begin{array}{rrr} 1 & 0 & 9 \\ 0 & 1 & -3 \end{array}\right] $$

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$ \left[\begin{array}{rrrrr} 3 & 2 & 4 & -1 & 5 \\ -2 & 0 & 1 & 3 & 2 \\ 1 & 0 & 0 & 4 & 0 \\ 6 & 0 & 2 & -1 & 0 \\ 3 & 0 & 5 & 1 & 0 \end{array}\right] $$

A corporation that makes sunglasses has four factories, each of which manufactures two products. The number of units of product \(i\) produced at factory \(j\) in one day is represented by \(a_{i j}\) in the matrix \(A=\left[\begin{array}{rrrr}100 & 120 & 60 & 40 \\ 140 & 160 & 200 & 80\end{array}\right] .\) Find the production levels if production is increased by \(10 \%\). (Hint: Because an increase of \(10 \%\) corresponds to \(100 \%+10 \%\), multiply the matrix by \(1.10 .\))

An augmented matrix that represents a system of linear equations (in variables \(x, y\), and \(z\) ) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. $$ \left[\begin{array}{rrrrr} 1 & 0 & 0 & -4 \\ 0 & 1 & 0 & -8 \\ 0 & 0 & 1 & 2 \end{array}\right] $$

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$ \left[\begin{array}{llllr} 5 & 2 & 0 & 0 & -2 \\ 0 & 1 & 4 & 3 & 2 \\ 0 & 0 & 2 & 6 & 3 \\ 0 & 0 & 3 & 4 & 1 \\ 0 & 0 & 0 & 0 & 2 \end{array}\right] $$

A tire corporation has three factories, each of which manufactures two products. The number of units of product \(i\) produced at factory \(j\) in one day is represented by \(a_{i j}\) in the matrix \(A=\left[\begin{array}{rrr}80 & 120 & 140 \\ 40 & 100 & 80\end{array}\right]\) Find the production levels if production is decreased by 5\%. (Hint: Because a decrease of \(5 \%\) corresponds to \(100 \%-5 \%\), multiply the matrix by \(0.95 .\) )

An augmented matrix that represents a system of linear equations (in variables \(x, y\), and \(z\) ) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. $$ \left[\begin{array}{rrrrr} 1 & 0 & 0 & \vdots & 3 \\ 0 & 1 & 0 & \vdots & -1 \\ 0 & 0 & 1 & 0 & 0 \end{array}\right] $$

Use the matrix capabilities of a graphing utility to evaluate the determinant. $$ \left|\begin{array}{rrr} 3 & 8 & -7 \\ 0 & -5 & 4 \\ 8 & 1 & 6 \end{array}\right| $$

An augmented matrix that represents a system of linear equations (in variables \(x, y\), and \(z\) ) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. $$ \left[\begin{array}{rrrrr} 1 & 0 & 2 & \vdots & -4 \\ 0 & 1 & 1 & \vdots & 6 \\ 0 & 0 & 0 & \vdots & 0 \end{array}\right] $$

Use the matrix capabilities of a graphing utility to evaluate the determinant. $$ \left|\begin{array}{rrr} 5 & -8 & 0 \\ 9 & 7 & 4 \\ -8 & 7 & 1 \end{array}\right| $$

A vacation service has identified four resort hotels with a special all- inclusive package (room and meals included) at a popular travel destination. The quoted room rates are for double and family (maximum of four people) occupancy for 5 days and 4 nights. The current rates for the two types of rooms at the four hotels are represented by the matrix \(A\). $$A=\left[\begin{array}{rrrr}615 & 670 & 740 & 990 \\ 995 & 1030 & 1180 & 1105\end{array}\right]$$ If room rates are guaranteed not to increase by more than \(12 \%\) by next season, what is the maximum rate per package per hotel?