Exam 7: Matrices and Determinants

Given $A = \left[ \begin{array} { c c } 1 & 2 \\- 3 & - 1\end{array} \right]$ , use $A ^ { - 1 }$ to decode the following cryptogram: -38 -1 -1 23 -69 -23 -51 -2 8 16 -4 32 5 10 -4 37 2 24

Use the matrix capabilities of a graphing utility to evaluate the expression. $- 2 \left( \left[ \begin{array} { c c } 8.8 & - 3.7 \\5.79 & - 4\end{array} \right] - \left[ \begin{array} { c c } - 4.49 & - 1.98 \\3.14 & - 7.63\end{array} \right] \right)$

Use a determinant to determine whether the points $( - 6,1 ) , ( - 8 , - 1 ) \text { and } ( - 3,3 )$ are collinear. Show all work.

Given $A = \left[ \begin{array} { c c c } 5 & - 10 & - 5 \\- 5 & 0 & 5 \\- 15 & 10 & 10\end{array} \right]$ , use $A ^ { - 1 }$ to decode the following cryptogram: -100 -70 35 -245 70 185 -140 40 115

You invest in AAA-rated bonds, A-rated bonds, and B-rated bonds. Your average yield is 8% on AAA bonds, 9% on A bonds, and 10% on B bonds. You invest twice as much in B bonds as in A bonds. The desired system of linear equations (where $x,$ $y,$ and $z$ represent the amounts invested in AAA, A, and B bonds, respectively) is as follows. $\left\{ \begin{aligned}x + y + z & = ( \text { total investment) } \\0.08 x + 0.09 y + 0.1 z & = \text { (annual return) } \\2 y - z & = 0\end{aligned} \right.$ Use the inverse of the coefficient matrix of this system to find the amount invested in B bonds for the given a total investment of $18,000 and annual return of $1640.

Write a cryptogram for the message "MERRY CHRISTMAS" using the matrix $\left[ \begin{array} { c c c } 0 & 2 & 1 \\- 4 & 1 & 4 \\4 & - 2 & 0\end{array} \right]$ . Show all your work.

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $\left\{ \begin{array} { l } 5 x + 4 y = - 9 \\x + 6 y = - 33\end{array} \right.$

Find which of the following matrices is an inverse to the matrix A below if any. $A = \left[ \begin{array} { c c } - 4 & - 3 \\- 1 & 7\end{array} \right]$ Matrix I: $\frac { 1 } { 31 } \left[ \begin{array} { c c } - 7 & - 3 \\ - 1 & 4 \end{array} \right]$ Matrix II : $\frac { 1 } { 31 } \left[ \begin{array} { c c } 4 & 3 \\ 1 & - 7 \end{array} \right]$ Matrix III : $\frac { 1 } { 31 } \left[ \begin{array} { c c } 7 & - 3 \\ - 1 & - 4 \end{array} \right]$

Determine a positive value for y such that a triangle with vertices $( - 8,4 ) , ( 7,2 )$ , and $( 0 , y )$ has an area of $23$ square units.

Write the augmented matrix of the system of equations below. $\left\{ \begin{aligned}8 x + 4 y + 6 z & = - 9 \\- 8 x - 4 y + 5 z & = - 1 \\- 4 x + 8 y - 2 z & = 8\end{aligned} \right.$

Use elementary row operations to write the matrix below in row echelon form. $\left[ \begin{array} { c c c c } 1 & 2 & 1 & - 9 \\3 & 2 & 3 & - 27 \\2 & 0 & 4 & - 26\end{array} \right]$

Identify the elementary row operation being performed to obtain the new row-equivalent matrix. Original $\mathrm { Mat } \mathrm { rix }~~~~~~~~~~$ New Row-Equivalent Matrix $\left[ \begin{array} { c c c } 6 & - 7 & - 6 \\7 & 6 & - 1\end{array} \right]~~~~~$$\left[ \begin{array} { c c c } 20 & 5 & - 8 \\7 & 6 & - 1\end{array} \right]$

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $\left\{ \begin{aligned}x - 5 y + 2 z & = - 1 \\- 6 x + 29 y - z & = 1\end{aligned} \right.$

Find A2. (Note: A2 = AA.) $A = \left[ \begin{array} { c } - 5 \\8\end{array} \right]$