Exam 9: Matrices and Determinants

Solve the system using the inverse that is given for the coefficient matrix. - $x + 2 y + 3 z = 6$ $x + y + z = - 8$ $x \quad - 2 z = 12$ The inverse of $\left[ \begin{array} { l l l } 1 & 2 & 3 \\ 1 & 1 & 1 \\ 1 & 0 & - 2 \end{array} \right]$ is $\left[ \begin{array} { r r r } - 2 & 4 & - 1 \\ 3 & - 5 & 2 \\ - 1 & 2 & - 1 \end{array} \right]$ .

Solve Problems Involving Systems Without Unique Solutions -A company that manufactures products A, B, and C does both assembly and testing. The hours needed to assemble and test each product are shown in the table below. Hours needed weekly to assemble Hours needed weekly to test Product A 1 4 Product B 1 5 Product C 2 10 The company has exactly 27 hours per week available for assembly and 120 hours per week available for testing. If the company must produce $t$ units of Product $C$ this week, how many units of Products $A$ and $B$ can they produce?

Determinants and Cramer's Rule 1 Evaluate a Second-Order Determinant - $\left| \begin{array} { c c } \frac { 1 } { 6 } & - \frac { 1 } { 10 } \\10 & 6\end{array} \right|$

Multiply Matrices - $A = \left[ \begin{array} { r r r } 1 & 3 & - 1 \\2 & 0 & 3\end{array} \right] , B = \left[ \begin{array} { r r } 3 & 0 \\- 1 & 1 \\0 & 3\end{array} \right]$

Add and Subtract Matrices -Let $\mathrm { A } = \left[ \begin{array} { r r } - 1 & 0 \\ 3 & 1 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } - 1 & 3 \\ 3 & 1 \end{array} \right]$ . Find $\mathrm { A } - \mathrm { B }$ .

Solve the system using the inverse that is given for the coefficient matrix. - $x + 2 y + 3 z = 12$ $x + y + z = - 5$ $2 x + 2 y + z = 1$$\quad$ The inverse of $\left[ \begin{array} { l l l } 1 & 2 & 3 \\ 1 & 1 & 1 \\ 2 & 2 & 1 \end{array} \right]$ is $\left[ \begin{array} { r r r } - 1 & 4 & - 1 \\ 1 & - 5 & 2 \\ 0 & 2 & - 1 \end{array} \right]$

Perform Scalar Multiplication -Let $\mathrm { A } = \left[ \begin{array} { r r } - 3 & 6 \\ 0 & 2 \end{array} \right]$ . Find 4A.

Multiply Matrices - $A = \left[ \begin{array} { r r r } 3 & - 2 & 1 \\ 0 & 4 & - 3 \end{array} \right] , B = \left[ \begin{array} { r r } 3 & 0 \\ - 2 & 1 \end{array} \right]$

Add and Subtract Matrices -Let $\mathrm { A } = \left[ \begin{array} { r r } - 5 & 1 \\ 2 & 5 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } 6 & 2 \\ 4 & - 2 \end{array} \right]$ . Find $\mathrm { A } + \mathrm { B }$ .

Evaluate a Third-Order Determinant -Determinants are used to show that three points lie on the same line (are collinear). If $\left| \begin{array} { l l l } x _ { 1 } & y _ { 1 } & 1 \\x _ { 2 } & y _ { 2 } & 1 \\x _ { 3 } & y _ { 3 } & 1\end{array} \right| = 0 ,$ then the points $\left( x _ { 1 } , y _ { 1 } \right) , \left( x _ { 2 } , y _ { 2 } \right)$ , and $\left( x _ { 3 } , y _ { 3 } \right)$ are collinear. If the determinant does not equal 0 , then the points are not collinear. Are the points $( - 10,5 ) , ( 0 , - 6 )$ , and $( - 20,18 )$ collinear?

Multiplicative Inverses of Matrices and Matrix Equations 1 Find the Multiplicative Inverse of a Square Matrix - $A = \left[ \begin{array} { r r } - 5 & - 1 \\ 6 & 0 \end{array} \right] , \quad B = \left[ \begin{array} { c c } 0 & \frac { 1 } { 6 } \\ - 1 & \frac { 5 } { 6 } \end{array} \right]$

Use Matrices and Gaussian Elimination to Solve Systems - x-y+4z =15 2x+z =3 x+2y+z =-3