Chapter 7

Explain why we can always evaluate the determinant of a square 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 any system of linear equations be written as an augmented matrix? Explain why or why not. Explain how to write that augmented matrix.

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.

Explain whether a system of two nonlinear equations can have exactly two solutions. What about exactly three? If not, explain why not. If so, give an example of such a system, in graph form, and explain why your choice gives two or three answers.

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

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.

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

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

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? (Hint: Think about it as a system of equations.)

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

If you are performing a breakeven 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.

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

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

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?

The determinant of \(2 \times 2\) matrix \(A\) is \(3 .\) If you switch the rows and multiply the first row by 6 and the second row by 2 , explain how to find the determinant and provide the answer.

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

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

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

If you are solving a break-even analysis and there is no breakeven 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 with zeros on the diagonal have an inverse? If so, find an example. If not, prove why not. For simplicity, assume a \(2 \times 2\) matrix.

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

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

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.

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

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

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

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

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. $$ \begin{array}{c} x+y=4 \\ x^{2}+y^{2}=9 \end{array} $$

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. $$ \left|\begin{array}{rr} 2 & -5 \\ -1 & 6 \end{array}\right| $$

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

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

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

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors. $$ \frac{3 x-79}{x^{2}-5 x-24} $$

For the following exercises, solve the system of nonlinear equations using substitution. $$ \begin{array}{c} y=x-3 \\ x^{2}+y^{2}=9 \end{array} $$

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

For the following exercises, find the determinant. $$ \left|\begin{array}{ll} -8 & 4 \\ -1 & 5 \end{array}\right| $$

In the following exercises, show that matrix \(A\) is the inverse of matrix \(B\). $$ A=\left[\begin{array}{ll} 4 & 5 \\ 7 & 0 \end{array}\right], \quad B=\left[\begin{array}{cc} 0 & \frac{1}{7} \\ \frac{1}{5} & -\frac{4}{35} \end{array}\right] $$

For the following exercises, write the augmented matrix for the linear system. $$ \begin{array}{c} 3 x+2 y+10 z=3 \\ -6 x+2 y+5 z=13 \\ 4 x+z=18 \end{array} $$

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

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

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

For the following exercises, determine whether the ordered triple given is the solution to the system of equations. $$ \begin{aligned} 6 x-7 y+z=& 2 \\ -x-y+3 z &=4 \quad \text { and }(4,2,-6) \\ 2 x+y-z &=1 \end{aligned} $$