Chapter 8

Operations with Matrices Find, if possible, (a) \(A B,\) (b) \(B A,\) and (c) \(A^{2}\). (Note: \(A^{2}=A A\).) Use the matrix capabilities of a graphing utility to verify your results. $$A=\left[\begin{array}{r} 7 \\ 8 \\ -1 \end{array}\right], \quad B=\left[\begin{array}{lll} 1 & 1 & 2 \end{array}\right]$$

Find a system of linear equations that has the given solution. (There are many correct answers.) $$\left(-6,-\frac{1}{2},-\frac{7}{4}\right)$$

Use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form . Use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form. $$\left[\begin{array}{ccc} 1 & 3 & 2 \\ 5 & 15 & 9 \\ 2 & 6 & 10 \end{array}\right]$$

Evaluate the determinants to verify the equation. $$\left|\begin{array}{ll} w & c x \\ y & c z \end{array}\right|=c\left|\begin{array}{ll} w & x \\ y & z \end{array}\right|$$

Use an inverse matrix to solve (if possible) the system of linear equations. $$\left\\{\begin{array}{l} 18 x+12 y=13 \\ 30 x+24 y=23 \end{array}\right.$$

Solve the system by the method of elimination and check any solutions using a graphing utility. \(\left\\{\begin{array}{l}\frac{2}{x}-\frac{1}{y}=5 \\\ \frac{6}{x}+\frac{1}{y}=11\end{array}\right.\)

Solve the system graphically. Verify your solutions algebraically. $$\left\\{\begin{aligned} y^{2}-x^{2}+9 &=0 \\ -\frac{1}{2} x+y &=\frac{3}{2} \end{aligned}\right.$$

Operations with Matrices Find, if possible, (a) \(A B,\) (b) \(B A,\) and (c) \(A^{2}\). (Note: \(A^{2}=A A\).) Use the matrix capabilities of a graphing utility to verify your results. $$A=\left[\begin{array}{llll} 3 & 2 & 1 \end{array}\right], \quad B=\left[\begin{array}{l} 2 \\ 3 \\ 0 \end{array}\right]$$

Find a system of linear equations that has the given solution. (There are many correct answers.) $$\left(-\frac{3}{2}, 4,-7\right)$$

Use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form . Use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form. $$\left[\begin{array}{rrrr} -4 & 1 & 0 & 6 \\ 1 & -2 & 3 & -4 \end{array}\right]$$

Evaluate the determinants to verify the equation. $$\left|\begin{array}{ll} w & x \\ y & z \end{array}\right|=\left|\begin{array}{ll} w & x+c w \\ y & z+c y \end{array}\right|$$

Use an inverse matrix to solve (if possible) the system of linear equations. $$\left\\{\begin{array}{r} -0.4 x+0.8 y=1.6 \\ 2 x-4 y=5 \end{array}\right.$$

Use a graphing utility to approximate all points of intersection of the graphs of equations in the system. Round your results to three decimal places. Verify your solutions by checking them in the original system. $$\left\\{\begin{array}{c} 7 x+8 y=24 \\ x-8 y=8 \end{array}\right.$$

Matrix Multiplication Use the matrix capabilities of a graphing utility to find \(A B,\) if possible. $$A=\left[\begin{array}{rrr} 1 & -12 & 4 \\ 14 & 10 & 12 \\ 6 & -15 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 12 & 10 \\ -6 & 12 \\ 10 & 16 \end{array}\right]$$

Find a system of linear equations that has the given solution. (There are many correct answers.) \((a, a+4, a),\) where \(a\) is a real number

Use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form . Use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form. $$\left[\begin{array}{rrrr} 5 & 1 & 2 & 4 \\ -1 & 5 & 10 & -32 \end{array}\right]$$

Evaluate the determinants to verify the equation. $$\left|\begin{array}{cc}w & x \\\c w & c x\end{array}\right|=0$$

Use an inverse matrix to solve (if possible) the system of linear equations. $$\left\\{\begin{array}{c} 0.2 x-0.6 y=2.4 \\ -x+1.4 y=-8.8 \end{array}\right.$$

Use a graphing utility to graph the lines in the system. Use the graphs to determine whether the system is consistent or inconsistent. If the system is consistent, determine the solution. Verify your results algebraically. $$\left\\{\begin{array}{c} -x+3 y=0 \\ 3 x-9 y=14 \end{array}\right.$$

Use a graphing utility to approximate all points of intersection of the graphs of equations in the system. Round your results to three decimal places. Verify your solutions by checking them in the original system. $$\left\\{\begin{aligned} x-y &=0 \\ 5 x-2 y &=6 \end{aligned}\right.$$

Matrix Multiplication Use the matrix capabilities of a graphing utility to find \(A B,\) if possible. $$A=\left[\begin{array}{rrrr} -3 & 8 & -6 & 8 \\ -12 & 15 & 9 & 6 \\ 5 & -1 & 1 & 5 \end{array}\right], \quad B=\left[\begin{array}{rrr} 3 & 1 & 6 \\ 24 & 15 & 14 \\ 16 & 10 & 21 \\ 8 & -4 & 10 \end{array}\right]$$

Find a system of linear equations that has the given solution. (There are many correct answers.) \((3 a, a, a+2),\) where \(a\) is a real number

Evaluate the determinants to verify the equation. $$\left|\begin{array}{lll} 1 & x & x^{2} \\ 1 & y & y^{2} \\ 1 & z & z^{2} \end{array}\right|=(y-x)(z-x)(z-y)$$

Use an inverse matrix to solve (if possible) the system of linear equations. $$\left\\{\begin{aligned} -\frac{1}{4} x+\frac{3}{8} y &=-2 \\ \frac{3}{2} x+\frac{3}{4} y &=-12 \end{aligned}\right.$$

Use a graphing utility to graph the lines in the system. Use the graphs to determine whether the system is consistent or inconsistent. If the system is consistent, determine the solution. Verify your results algebraically. $$\left\\{\begin{aligned} 6 x+3 y &=-8 \\ -x-\frac{1}{2} y &=\frac{4}{3} \end{aligned}\right.$$

Use a graphing utility to approximate all points of intersection of the graphs of equations in the system. Round your results to three decimal places. Verify your solutions by checking them in the original system. $$\left\\{\begin{array}{l} x-y^{2}=-1 \\ x-y=5 \end{array}\right.$$

Matrix Multiplication Use the matrix capabilities of a graphing utility to find \(A B,\) if possible. $$A=\left[\begin{array}{rrrr} 7 & 6 & 9 & -4 \\ 3 & -4 & 11 & -2 \\ -5 & -8 & 1 & 12 \end{array}\right], \quad B=\left[\begin{array}{rr} 15 & 8 \\ 23 & -17 \\ 9 & 10 \end{array}\right]$$

Sketch the plane represented by the linear equation. Then list four points that lie in the plane. $$x+y+z=8$$

Write the system of linear equations represented by the augmented matrix. Then use back-substitution to find the solution. (Use the variables \(x, y,\) and \(z,\) if applicable.) $$\left[\begin{array}{llll} 1 & 8 & \vdots & 12 \\ 0 & 1 & \vdots & 3 \end{array}\right]$$

Evaluate the determinants to verify the equation. $$\left|\begin{array}{ccc}a+b & a & a \\\a & a+b & a \\\a & a & a+b\end{array}\right|=b^{2}(3 a+b)$$

Use an inverse matrix to solve (if possible) the system of linear equations. $$\left\\{\begin{aligned} \frac{5}{6} x-y &=-10 \\ -\frac{5}{4} x+\frac{3}{2} y &=-2 \end{aligned}\right.$$

Use a graphing utility to graph the lines in the system. Use the graphs to determine whether the system is consistent or inconsistent. If the system is consistent, determine the solution. Verify your results algebraically. $$\left\\{\begin{aligned} -\frac{1}{4} x-\frac{1}{2} y &=1 \\ 5 x+y &=1 \end{aligned}\right.$$

Use a graphing utility to approximate all points of intersection of the graphs of equations in the system. Round your results to three decimal places. Verify your solutions by checking them in the original system. $$\left\\{\begin{array}{l} x-y^{2}=-2 \\ x-2 y=6 \end{array}\right.$$

Sketch the plane represented by the linear equation. Then list four points that lie in the plane. $$x+2 y+z=4$$

Write the system of linear equations represented by the augmented matrix. Then use back-substitution to find the solution. (Use the variables \(x, y,\) and \(z,\) if applicable.) $$\left[\begin{array}{rrrrr} 1 & -1 & 4 & \vdots & 0 \\ 0 & 1 & -1 & \vdots & 2 \\ 0 & 0 & 1 & \vdots & -2 \end{array}\right]$$

Solve for \(x\) $$\left|\begin{array}{ll} x & 2 \\ 1 & x \end{array}\right|=2$$

Use an inverse matrix to solve (if possible) the system of linear equations. $$\left\\{\begin{array}{l} 4 x-y+z=-5 \\ 2 x+2 y+3 z=10 \\ 5 x-2 y+6 z=1 \end{array}\right.$$

Use a graphing utility to graph the lines in the system. Use the graphs to determine whether the system is consistent or inconsistent. If the system is consistent, determine the solution. Verify your results algebraically. $$\left\\{\begin{aligned} 3.2 x-16 y &=7.5 \\ x-5 y &=-9 \end{aligned}\right.$$

Use a graphing utility to approximate all points of intersection of the graphs of equations in the system. Round your results to three decimal places. Verify your solutions by checking them in the original system. $$\left\\{\begin{aligned} x^{2}+y^{2} &=8 \\ y &=x^{2} \end{aligned}\right.$$

Operations with Matrices Use the matrix capabilities of a graphing utility to evaluate the expression. $$\left[\begin{array}{rr} 3 & 1 \\ 0 & -2 \end{array}\right]\left[\begin{array}{rr} 1 & 0 \\ -2 & 2 \end{array}\right]\left[\begin{array}{ll} 1 & 0 \\ 2 & 4 \end{array}\right]$$

Sketch the plane represented by the linear equation. Then list four points that lie in the plane. $$3 x+2 y+2 z=12$$

Write the system of linear equations represented by the augmented matrix. Then use back-substitution to find the solution. (Use the variables \(x, y,\) and \(z,\) if applicable.) $$\left[\begin{array}{ccccc} 1 & 0 & -2 & \vdots & -7 \\ 0 & 1 & 1 & \vdots & 9 \\ 0 & 0 & 1 & \vdots & -3 \end{array}\right]$$

Solve for \(x\) $$\left|\begin{array}{rr} x & 4 \\ -1 & x \end{array}\right|=20$$

Use an inverse matrix to solve (if possible) the system of linear equations. $$\left\\{\begin{array}{l} 4 x-2 y+3 z=-2 \\ 2 x+2 y+5 z=16 \\ 8 x-5 y-2 z=4 \end{array}\right.$$

Use a graphing utility to graph the lines in the system. Use the graphs to determine whether the system is consistent or inconsistent. If the system is consistent, determine the solution. Verify your results algebraically. $$\underline{\phantom{xxx}}\left\\{\begin{array}{l} -6 x+4 y=-9 \\ 4.5 x-3 y=6.75 \end{array}\right.$$

Use a graphing utility to approximate all points of intersection of the graphs of equations in the system. Round your results to three decimal places. Verify your solutions by checking them in the original system. $$\left\\{\begin{aligned} x^{2}+y^{2} &=25 \\ (x-8)^{2}+y^{2} &=41 \end{aligned}\right.$$

Operations with Matrices Use the matrix capabilities of a graphing utility to evaluate the expression. $$\left[\begin{array}{rrr} 6 & 5 & -1 \\ 1 & -2 & 0 \end{array}\right]\left[\begin{array}{rr} 0 & 3 \\ -1 & -3 \\ 4 & 1 \end{array}\right]\left[\begin{array}{rr} -2 & 2 \\ 0 & -1 \end{array}\right]$$

Sketch the plane represented by the linear equation. Then list four points that lie in the plane. $$5 x+y+3 z=15$$

An augmented matrix that represents a system of linear equations (in the variables \(x\) and \(y\) or \(x, y,\) and \(z\) ) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. $$\left[\begin{array}{cccc} 1 & 0 & \vdots & 7 \\ 0 & 1 & \vdots & -5 \end{array}\right]$$

Solve for \(x\) $$\left|\begin{array}{cc} 2 x & -3 \\ -2 & 2 x \end{array}\right|=3$$