Exam 49: The Inverse of a Square Matrix

A small home business creates muffins,bones,and cookies for dogs.In addition to other ingredients,each muffin requires 2 units of beef,3 units of chicken,and 2 units of liver.Each bone requires 1 unit of beef,1 unit of chicken,and 1 unit of liver.Each cookie requires 2 units of beef,1 unit of chicken,and 1.5 units of liver.Find the numbers of muffins,bones,and cookies that the company can create with the given amounts of ingredients. ​ 875 units of beef 830 units of chicken 850 units of liver ​

​Solve the system of linear equations​ $\left\{ \begin{array} { l l } 7 x + 7 y + 7 z & = 0 \\21 x + 35 y + 28 z & = 6 \\21 x + 42 y + 35 z & = 1\end{array} \right.$ ​using the inverse matrix $\frac { 1 } { 7 } \left[ \begin{array} { c c c } 1 & 1 & - 1 \\- 3 & 2 & - 1 \\3 & - 3 & 2\end{array} \right]$ .

A small home business creates muffins,bones,and cookies for dogs.In addition to other ingredients,each muffin requires 2 units of beef,3 units of chicken,and 2 units of liver.Each bone requires 1 unit of beef,1 unit of chicken,and 1 unit of liver.Each cookie requires 2 units of beef,1 unit of chicken,and 1.5 units of liver.Find the numbers of muffins,bones,and cookies that the company can create with the given amounts of ingredients. ​ 800 units of beef 750 units of chicken 725 units of liver ​

Use the inverse formula $A ^ { - 1 } = \frac { 1 } { a d - b c } \left[ \begin{array} { c c } d & - b \\- c & a\end{array} \right]$ to find the inverse of the 2×2 matrix (if it exists). ​​ $\left[ \begin{array} { c c } - 12 & 3 \\5 & 2\end{array} \right]$ ​

Find the inverse of the following matrix.​ $\left[ \begin{array} { c c } \cos \theta & \sin \theta \\- \sin \theta & \cos \theta\end{array} \right]$ ​

A coffee manufacturer sells a 14-pound package that contains three flavors of coffee for $25.French vanilla coffee costs $5 per pound,hazelnut flavored coffee costs $5.50 per pound,and Swiss chocolate flavored coffee costs $6 per pound.The package contains the same amount of hazelnut as Swiss chocolate.Let f represent the number of pounds of French vanilla,h represent the number of pounds of hazelnut,and s represent the number of pounds of Swiss chocolate. ​ Write a system of linear equations that represents the situation. ​

Use an inverse matrix to solve (if possible)the system of linear equations.​ $\left\{ \begin{array} { l } 0.2 x - 0.6 y = 14.4 \\- x + 1.4 y = - 20.8\end{array} \right.$ ​

Consider a person who invests in AAA-rated bonds,A-rated bonds,and B-rated bonds.The average yields are 6.5% on AAA bonds,7% on A bonds,and 9% on B bonds.The person invests twice as much in B bonds as in A bonds.Let x,y and z represent the amounts invested in AAA,A,and B bonds,respectively. Total Investment Annual Return $12,000 890 $\left\{ \begin{array} { c l } x + y + z & = 12,000 \text { (total investment) } \\0.065 x + 0.07 y + 0.09 z & = 890 \text { (annual retum) } \\2 y - z & = 0\end{array} \right.$ Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond.

Find the inverse of the matrix $\left[ \begin{array} { c c } - 1 & 4 \\3 & - 2\end{array} \right]$ (if it exists).

A small home business creates muffins,bones,and cookies for dogs.In addition to other ingredients,each muffin requires 2 units of beef,3 units of chicken,and 2 units of liver.Each bone requires 1 unit of beef,1 unit of chicken,and 1 unit of liver.Each cookie requires 2 units of beef,1 unit of chicken,and 1.5 units of liver.Find the numbers of muffins,bones,and cookies that the company can create with the given amounts of ingredients. ​ 900 units of beef 700 units of chicken 800 units of liver ​

Solve the system of linear equations.​ $\left\{ \begin{array} { l } x + y + z = 0 \\3 x + 5 y + 4 z = 9 \\3 x + 6 y + 5 z = 6\end{array} \right.$ ​

Find the inverse of the matrix.​ $\left[ \begin{array} { c c c } 1 & 0 & 5 \\1 & 1 & 5 \\- 5 & 1 & 1\end{array} \right]$ ​

Find the inverse of the matrix.​ $\left[ \begin{array} { l l } 4 & 1 \\1 & 4\end{array} \right]$ ​

Use a graphing calculator to find the inverse of the matrix.​ $\left[ \begin{array} { l l l l } 1 & 4 & 8 & 3 \\0 & 1 & 4 & 8 \\0 & 0 & 1 & 4 \\0 & 0 & 0 & 1\end{array} \right]$ ​ ​

Find the inverse of the matrix $\left[ \begin{array} { c c } 2 & 4 \\- 6 & - 10\end{array} \right]$ .

Find the inverse of the matrix (if it exists). ​​ $\left[ \begin{array} { l l } 1 & 2 \\3 & 7\end{array} \right]$ ​

A florist is creating 10 centerpieces for the tables at a wedding reception.Roses cost $2.50 each,lilies cost $8 each,and irises cost $4 each.The customer has a budget of $300 allocated for the centerpieces and wants each centerpiece to contain 12 flowers,with twice as many roses as the number of irises and lilies combined. ​ Write a system of linear equations that represents the situation. ​

​Use the matrix capabilities of a graphing utility to solve the following system of linear equations:​ $\left\{ \begin{array} { l l } - 14 x + 14 y + 21 z & = 10 \\7 x - 7 y & = - 5 \\7 x + 28 z & = 20\end{array} \right.$ ​

Find the inverse of the matrix (if it exists).​ $\left[ \begin{array} { c c c } 4 & 0 & 0 \\6 & 0 & 0 \\5 & 11 & 11\end{array} \right]$ ​