Chapter 7

Explain why we can always evaluate the determinant of a square matrix.

Can any system of linear equations be written as an augmented matrix? Explain why or why not. Explain how to write that augmented matrix.

In a previous section, we showed that matrix multiplication is not commutative, that is, \(A B \neq B A\) in most cases. Can you explain why matrix multiplication is commutative for matrix inverses, that is, \(A^{-1} A=A A^{-1}\) ?

Can we add any two matrices together? If so, explain why; if not, explain why not and give an example of two matrices that cannot be added together.

Can a linear system of three equations have exactly two solutions? Explain why or why not?

Can any quotient of polynomials be decomposed into at least two partial fractions? If so, explain why, and if not, give an example of such a fraction.

Can a system of linear equations have exactly two solutions? Explain why or why not.

Examining Cramer's Rule, explain why there is no unique solution to the system when the determinant of your matrix is \(0 .\) For simplicity, use a \(2 \times 2\) matrix.

Can any matrix be written as a system of linear equations? Explain why or why not. Explain how to write that system of equations.

Does every \(2 \times 2\) matrix have an inverse? Explain why or why not. Explain what condition is necessary for an inverse to exist.

When graphing an inequality, explain why we only need to test one point to determine whether an entire region is the solution?

Can we multiply any column matrix by any row matrix? Explain why or why not.

Can you explain why a partial fraction decomposition is unique?

If you are performing a break-even analysis for a business and their cost and revenue equations are dependent, explain what this means for the company's profit margins.

Explain what it means in terms of an inverse for a matrix to have a 0 determinant.

Is there only one correct method of using row operations on a matrix? Try to explain two different row operations possible to solve the augmented matrix \(\left[\begin{array}{rr|r}9 & 3 & 0 \\ 1 & -2 & 6\end{array}\right]\)..

Can you explain whether a \(2 \times 2\) matrix with an entire row of zeros can have an inverse?

When you graph a system of inequalities, will there always be a feasible region? If so, explain why. If not, give an example of a graph of inequalities that does not have a feasible region. Why does it not have a feasible region?

Can you explain how to verify a partial fraction decomposition graphically?

If you are solving a break-even analysis and get a negative break-even point, explain what this signifies or the company?

The determinant of \(2 \times 2\) matrix \(A\) is \(3 .\) If you switch the rows and multiply the fi st row by 6 and the second row by 2 , explain how to fi \(\mathrm{d}\) the determinant and provide the answer.

Can a matrix with an entire column of zeros have an inverse? Explain why or why not.

If you graph a revenue and cost function, explain how to determine in what regions there is profit.

Can any two matrices of the same size be multiplied? If so, explain why, and if not, explain why not and give an example of two matrices of the same size that cannot be multiplied together.

You are unsure if you correctly decomposed the partial fraction correctly. Explain how you could double check your answer.

If you are solving a break-even analysis and there is no break-even point, explain what this means for the company. How should they ensure there is a break-even point?

For the following exercises, find the determinant. \(\left|\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right|\)

Can a matrix that has 0 entries for an entire row have one solution? Explain why or why not.

Can a matrix with zeros on the diagonal have an inverse? If so, fi d an example. If not, prove why not. For simplicity, assume a \(2 \times 2\) matrix.

Does matrix multiplication commute? That is, does \(A B=B A ?\) If so, prove why it does. If not, explain why it does not.

Can you explain whether there can be only one method to solve a linear system of equations? If yes, give an example of such a system of equations. If not, explain why not.

Once you have a system of equations generated by the partial fraction decomposition, can you explain another method to solve it? For example if you had \(\frac{7 x+13}{3 x^{2}+8 x+15}=\frac{A}{x+1}+\frac{B}{3 x+5},\) we eventually simplify to \(7 x+13=A(3 x+5)+B(x+1)\) Explain how you could intelligently choose an \(x\) -value that will eliminate either \(A\) or \(B\) and solve for \(A\) and \(B\).

For the following exercises, find the determinant. \(\left|\begin{array}{rr}-1 & 2 \\ 3 & -4\end{array}\right|\)

Write the augmented matrix for the linear system. \(\begin{aligned} 8 x-37 y &=8 \\ 2 x+12 y &=3 \end{aligned}\)

In the following exercises, show that matrix \(A\) is the inverse of matrix \(B\). $$A=\left[\begin{array}{rr}1 & 0 \\ -1 & 1\end{array}\right], B=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]$$

Solve the system of nonlinear equations using substitution. $$ \begin{array}{r} x+y=4 \\ x^{2}+y^{2}=9 \end{array} $$

For the following exercises, write the augmented matrix for the linear system. $$ \begin{array}{l}{8 x-37 y=8} \\ {2 x+12 y=3}\end{array} $$

Use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined. \(A=\left[\begin{array}{ll}1 & 3 \\ 0 & 7\end{array}\right], B=\left[\begin{array}{rr}2 & 14 \\ 22 & 6\end{array}\right], C=\left[\begin{array}{rr}1 & 5 \\ 8 & 92 \\ 12 & 6\end{array}\right], D=\left[\begin{array}{rr}10 & 14 \\ 7 & 2 \\ 5 & 61\end{array}\right], E=\left[\begin{array}{rr}6 & 12 \\ 14 & 5\end{array}\right], F=\left[\begin{array}{rr}0 & 9 \\ 78 & 17 \\ 15 & 4\end{array}\right]\) \(A+B\)

Determine whether the ordered triple given is the solution to the system of equations. $$ \begin{aligned} 2 x-6 y+6 z &=-12 \\ x+4 y+5 z &=-1 \quad \text { and }(0,1,-1) \\ -x+2 y+3 z &=-1 \end{aligned} $$

For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined. $$ A=\left[\begin{array}{ll}{1} & {3} \\ {0} & {7}\end{array}\right], B=\left[\begin{array}{cc}{2} & {14} \\ {22} & {6}\end{array}\right], C=\left[\begin{array}{cc}{1} & {5} \\ {8} & {92} \\ {12} & {6}\end{array}\right], D=\left[\begin{array}{cc}{10} & {14} \\ {7} & {2} \\\ {5} & {61}\end{array}\right], E=\left[\begin{array}{cc}{6} & {12} \\ {14} & {5}\end{array}\right], F=\left[\begin{array}{cc}{0} & {9} \\ {78} & {17} \\\ {15} & {4}\end{array}\right] $$ $$ A+B $$

Find the decomposition of the partial fraction for the nonrepeating linear factors. \(\frac{5 x+16}{x^{2}+10 x+24}\)

Determine whether the given ordered pair is a solution to the system of equations. $$ \begin{array}{l} 5 x-y=4 \\ x+6 y=2 \text { and }(4,0) \end{array} $$

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors. $$\frac{5 x+16}{x^{2}+10 x+24}$$

For the following exercises, solve the system of nonlinear equations using substitution. $$\begin{aligned} x+y &=4 \\ x^{2}+y^{2} &=9 \end{aligned}$$

For the following exercises, determine whether the ordered triple given is the solution to the system of equations. $$ \begin{aligned} 2 x-6 y+6 z &=-12 \\ x+4 y+5 z &=-1 \quad \text { and }(0,1,-1) \\\\-x+2 y+3 z &=-1 \end{aligned} $$

For the following exercises, find the determinant. \(\left|\begin{array}{rr}2 & -5 \\ -1 & 6\end{array}\right|\)

Write the augmented matrix for the linear system. \(\begin{aligned} 16 y &=4 \\ 9 x-y &=2 \end{aligned}\)

In the following exercises, show that matrix \(A\) is the inverse of matrix \(B\). $$A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right], B=\left[\begin{array}{rr}-2 & 1 \\ \frac{3}{2} & -\frac{1}{2}\end{array}\right]$$

Solve the system of nonlinear equations using substitution. $$ \begin{aligned} y &=x-3 \\ x^{2}+y^{2} &=9 \end{aligned} $$

For the following exercises, write the augmented matrix for the linear system. $$ \begin{aligned} 16 y &=4 \\ 9 x-y &=2 \end{aligned} $$