Chapter 9

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Matrix row operations remind me of what I did when solving a linear system by the addition method, although I no longer write the variables.

Without going into too much detail, describe how to solve a linear system in three variables using Cramer's Rule.

Explain how to find the multiplicative inverse for a \(2 \times 2\) invertible matrix.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When I use matrices to solve linear systems, the only arithmetic involves multiplication or a combination of multiplication and addition.

In applying Cramer's Rule, what should you do if \(D=0 ?\)

Explain how to write a linear system of three equations in three variables as a matrix equation.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When I use matrices to solve linear systems, I spend most of my time using row operations to express the system's augmented matrix in row-echelon form.

The process of solving a linear system in three variables using Cramer's Rule can involve tedious computation. Is there a way of speeding up this process, perhaps using Cramer's Rule to find the value for only one of the variables? Describe how this process might work, presenting a specific example with your description. Remember that your goal is still to find the value for each variable in the system.

What is a cryptogram?

Explain how to solve the matrix equation \(A X=B\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Using row operations on an augmented matrix, I obtain a row in which 0 s appear to the left of the vertical bar, but 6 appears on the right, so the system I'm working with has no solution.

If you could use only one method to solve linear systems in three variables, which method would you select? Explain why this is so.

It's January \(1,\) and you've written down your major goal for the year. You do not want those closest to you to see what you've written in case you do not accomplish your objective. Consequently, you decide to use a coding matrix to encode your goal. Explain how this can be accomplished.

It's January \(1,\) and you've written down your major goal for the year. You do not want those closest to you to see what you've written in case you do not accomplish your objective. Consequently, you decide to use a coding matrix to encode your goal. Explain how this can be accomplished.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. A matrix row operation such as \(-\frac{4}{5} R_{1}+R_{2}\) is not permitted because of the negative fraction.

Use a graphing utility to evaluate the determinant for the given matrix. $$ \left[\begin{array}{rrrr} {3} & {-2} & {-1} & {4} \\ {-5} & {1} & {2} & {7} \\ {2} & {4} & {5} & {0} \\ {-1} & {3} & {-6} & {5} \end{array}\right] $$

What is meant by the order of a matrix? Give an example with your explanation.

Use a graphing utility to evaluate the determinant for the given matrix. $$ \left[\begin{array}{rrrrr} {8} & {2} & {6} & {-1} & {0} \\ {2} & {0} & {-3} & {4} & {7} \\ {2} & {1} & {-3} & {6} & {-5} \\ {-1} & {2} & {1} & {5} & {-1} \\ {4} & {5} & {-2} & {3} & {-8} \end{array}\right] $$

Use a graphing utility to find the multiplicative inverse of each matrix. Check that the displayed inverse is correct. $$ \left[\begin{array}{rr} {3} & {-1} \\ {-2} & {1} \end{array}\right] $$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The row operation \(k R_{i}+R_{j}\) indicates that it is the elements in row \(i\) that change.

Use a graphing utility to find the multiplicative inverse of each matrix. Check that the displayed inverse is correct. $$ \left[\begin{array}{rr} {-4} & {1} \\ {6} & {-2} \end{array}\right] $$

What is the fastest method for solving a linear system with your graphing utility?

What are equal matrices?

The table shows the daily production level and profit for a business. y (Daily Profit) $$ \begin{array}{ll} {x \text { (Number of Units }} & {30} & {50} & {100} \\ {\text { Produced Daily) }} \\ {y \text { (Daily Profit) }} & {\$ 5900} & {\$ 7500} & {\$ 4500} \end{array} $$ Use the quadratic function \(y=a x^{2}+b x+c\) to determine the number of units that should be produced each day for maximum profit. What is the maximum daily profit?

Use a graphing utility to find the multiplicative inverse of each matrix. Check that the displayed inverse is correct. $$ \left[\begin{array}{rrr} {-2} & {1} & {-1} \\ {-5} & {2} & {-1} \\ {3} & {-1} & {1} \end{array}\right] $$

Solve the system: \(\left\\{\begin{array}{l}{x-y=2} \\ {y^{2}=4 x+4}\end{array}\right.\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can speed up the tedious computations required by Cramer's Rule by using the value of \(D\) to determine the value of \(D_{x}\).

Use a graphing utility to find the multiplicative inverse of each matrix. Check that the displayed inverse is correct. $$ \left[\begin{array}{rrr} {1} & {1} & {-1} \\ {-3} & {2} & {-1} \\ {3} & {-3} & {2} \end{array}\right] $$

How are matrices added?

Describe how to subtract matrices.

Graph the solution set of the system: $$ \left\\{\begin{array}{l} {x+y \leq 7} \\ {x+4 y>-8} \end{array}\right. $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When using Cramer's Rule to solve a linear system, the number of determinants that I set up and evaluate is the same as the number of variables in the system.

Use a graphing utility to find the multiplicative inverse of each matrix. Check that the displayed inverse is correct. $$ \left[\begin{array}{rrrr} {7} & {-3} & {0} & {2} \\ {-2} & {1} & {0} & {-1} \\ {4} & {0} & {1} & {-2} \\ {-1} & {1} & {0} & {-1} \end{array}\right] $$

Describe matrices that cannot be added or subtracted.

Write as a single logarithm: $$ 3 \log _{b} x-2 \log _{b} 5-\frac{1}{3} \log _{b} y $$

Use a graphing utility to find the multiplicative inverse of each matrix. Check that the displayed inverse is correct. $$ \left[\begin{array}{llll} {1} & {2} & {0} & {0} \\ {0} & {0} & {1} & {0} \\ {1} & {3} & {0} & {1} \\ {4} & {0} & {0} & {2} \end{array}\right] $$

Describe how to perform scalar multiplication. Provide an example with your description.

Solve: \(2 \cos ^{2} x+3 \sin x-3=0, \quad 0 \leq x<2 \pi\)

a. Evaluate: \(\left|\begin{array}{ll}{a} & {a} \\ {0} & {a}\end{array}\right|\) b. Evaluate: \(\left|\begin{array}{ccc}{a} & {a} & {a} \\ {0} & {a} & {a} \\\ {0} & {0} & {a}\end{array}\right|\) c. Evaluate: \(:\left|\begin{array}{cccc}{a} & {a} & {a} & {a} \\ {0} & {a} & {a} & {a} \\ {0} & {0} & {a} & {a} \\ {0} & {0} & {0} & {a}\end{array}\right|\) d. Describe the pattern in the given determinants. e. Describe the pattern in the evaluations.

Write each system in the form \(A X=B .\) Then solve the system by entering \(A\) and \(B\) into your graphing utility and computing \(A^{-1} B\) $$ \left\\{\begin{aligned} x-y+z &=-6 \\ 4 x+2 y+z &=9 \\ 4 x-2 y+z &=-3 \end{aligned}\right. $$

Describe how to multiply matrices.

Exercises \(72-74\) will help you prepare for the material covered in the next section. In each exercise, refer to the following system: $$ \left\\{\begin{aligned} 3 x-4 y+4 z &=7 \\ x-y-2 z &=2 \\ 2 x-3 y+6 z &=5 \end{aligned}\right. $$ Show that \((12 z+1,10 z-1, z)\) satisfies the system for \(z=0\)

$$ \text { Evaluate: }:\left(\begin{array}{lllll} {2} & {0} & {0} & {0} & {0} \\ {0} & {3} & {0} & {0} & {0} \\ {0} & {0} & {2} & {0} & {0} \\ {0} & {0} & {0} & {1} & {0} \\ {0} & {0} & {0} & {0} & {4} \end{array}\right) $$

Write each system in the form \(A X=B .\) Then solve the system by entering \(A\) and \(B\) into your graphing utility and computing \(A^{-1} B\) $$ \left\\{\begin{aligned} y+2 z &=0 \\ -x+y &=1 \\ 2 x-y+z &=-1 \end{aligned}\right. $$

Describe when the multiplication of two matrices is not defined.

Exercises \(72-74\) will help you prepare for the material covered in the next section. In each exercise, refer to the following system: $$ \left\\{\begin{aligned} 3 x-4 y+4 z &=7 \\ x-y-2 z &=2 \\ 2 x-3 y+6 z &=5 \end{aligned}\right. $$ Show that \((12 z+1,10 z-1, z)\) satisfies the system for \(z=1\)

What happens to the value of a second-order determinant if the two columns are interchanged?

Write each system in the form \(A X=B .\) Then solve the system by entering \(A\) and \(B\) into your graphing utility and computing \(A^{-1} B\) $$ \left\\{\begin{array}{l} {3 x-2 y+z=-2} \\ {4 x-5 y+3 z=-9} \\ {2 x-y+5 z=-5} \end{array}\right. $$

If two matrices can be multiplied, describe how to determine the order of the product.

Consider the system $$ \left\\{\begin{array}{l} {a_{1} x+b_{1} y=c_{1}} \\ {a_{2} x+b_{2} y=c_{2}} \end{array}\right. $$ Use Cramer's Rule to prove that if the first equation of the system is replaced by the sum of the two equations, the resulting system has the same solution as the original system.