91Ó°ÊÓ

Testing for Collinear Points In Exercises \(35-40\) , use a determinant to determine whether the points are collinear. $$(3,-1),(0,-3),(12,5)$$

Testing for Collinear Points In Exercises \(35-40\) , use a determinant to determine whether the points are collinear. $$\left(2,-\frac{1}{2}\right),(-4,4),(6,-3)$$

Comparing Linear Systems and Matrix Operations In Exercises 41 and \(42,\) (a) perform the row operations to solve the augmented matrix, (b) write and solve the system of linear equations represented by the augmented matrix, and (c) compare the two solution methods. Which do you prefer? $$\left[ \begin{array}{rrrr}{-3} & {4} & {\vdots} & {22} \\ {6} & {-4} & {\vdots} & {-28}\end{array}\right]$$ $$\begin{array}{l}{\text { (i) Add } R_{2} \text { to } R_{1} \text { . }} \\\ {\text { (ii) Add }-2 \text { times } R_{1} \text { to } R_{2} \text { . }} \\\ {\text { (iii) Multiply } R_{2} \text { by }-\frac{1}{4}} \\ {\text { (iv) Multiply } R_{1} \text { by } \frac{1}{3}}\end{array}$$

Encoding a Message In Exercises 49 and 50 , (a) write the uncoded \(1 \times 2\) row matrices for the message, and then (b) encode the message using the encoding matrix. $$Message$$ $$. COME HOME SOON$$ $$Encoding Matrix$$ $$\left[ \begin{array}{ll}{1} & {2} \\ {3} & {5}\end{array}\right]$$

Finding the Determinant of a Matrix In Exercises \(47-62,\) find the determinant of the matrix. Expand by cofactors using the row or column that appears to make the computations easiest. $$\left[ \begin{array}{rrr}{1} & {1} & {2} \\ {3} & {1} & {0} \\ {-2} & {0} & {3}\end{array}\right]$$

Encoding a Message In Exercises \(53-56\) , write a cryptogram for the message using the matrix $$A=\left[ \begin{array}{rrr}{1} & {2} & {2} \\ {3} & {7} & {9} \\ {-1} & {-4} & {-7}\end{array}\right]$$ ICEBERG DEAD AHEAD

Using Back-Substitution, write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve. (Use variables \(x, y,\) and \(z,\) if applicable.) $$\left[ \begin{array}{rrrrr}{1} & {-1} & {2} & {\vdots} & {4} \\ {0} & {1} & {-1} & {\vdots} & {2} \\ {0} & {0} & {1} & {\vdots} & {-2}\end{array}\right]$$

Interpreting Reduced Row-Echelon Form , an augmented matrix that represents a system of linear equations (in variables \(x, y,\) and \(z,\) if applicable) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. $$\left[ \begin{array}{rrrr}{1} & {0} & {0} & {\vdots} & {-4} \\ {0} & {1} & {0} & {\vdots} & {-10} \\ {0} & {0} & {1} & {\vdots} & {4}\end{array}\right]$$

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system. $$\left\\{\begin{aligned} 2 x+10 y+2 z=& 6 \\ x+5 y+2 z=& 6 \\ x+5 y+z=& 3 \\\\-3 x-15 y-3 z=&-9 \end{aligned}\right.$$

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system. $$\left\\{\begin{aligned} x+2 y+2 z+4 w=& 11 \\ 3 x+6 y+5 z+12 w=& 30 \\ x+3 y-3 z+2 w=&-5 \\ 6 x-y-z+\quad w=&-9 \end{aligned}\right.$$