Lesson 5 Solving Systems with Matrices, Revisited Solidify Understanding

Ready

1.

The following three points form the vertices of a triangle: $(3, 2)$ , $(6, 1)$ , $(4, 3)$ .

Plot these three points on the coordinate grid and then connect them.
Reflect the original triangle over the $y$ -axis and record the coordinates of the vertices.
Reflect the original triangle over the $x$ -axis and record the coordinates of the vertices.
Rotate the original triangle $90 °$ counterclockwise about the origin and record the coordinates of the vertices.
Rotate the original triangle $180 °$ about the origin and record the coordinates of the vertices.

Set

2.

Two of the following systems have unique solutions (that is, the lines intersect at a single point).

Use the determinant of a $2 \times 2$ matrix to decide which systems have unique solutions, and which one does not.

a.

${\begin{cases} 8 x - 2 y = - 2 \\ 4 x + y = 5 \end{cases}$

b.

${\begin{cases} 3 x + 2 y = 7 \\ 6 x + 4 y = - 5 \end{cases}$

c.

${\begin{cases} 4 x + 2 y = 0 \\ 3 x + y = 2 \end{cases}$

3.

For each of the systems in problem 2 that have a unique solution, find the solution to the system by solving a matrix equation using an inverse matrix.

a.

${\begin{cases} 8 x - 2 y = - 2 \\ 4 x + y = 5 \end{cases}$

b.

${\begin{cases} 3 x + 2 y = 7 \\ 6 x + 4 y = - 5 \end{cases}$

c.

${\begin{cases} 4 x + 2 y = 0 \\ 3 x + y = 2 \end{cases}$

Go

Perform the indicated operation. If the operation cannot be performed, state why.

4.

$[\begin{array}{rrr} 2 & 0 & 5 \\ - 3 & 7 & 1 \end{array}] \cdot [\begin{array}{rr} - 1 & 4 \\ 8 & 6 \\ 0 & - 2 \end{array}]$

5.

$[\begin{array}{rrr} 2 & 0 & 5 \\ - 3 & 7 & 1 \end{array}] + [\begin{array}{rr} - 1 & 4 \\ 8 & 6 \\ 0 & - 2 \end{array}]$

6.

$[\begin{array}{rr} 12 & 34 \\ 9 & - 3 \end{array}] - [\begin{array}{rr} 8 & 14 \\ 2 & 7 \end{array}]$

7.

$[\begin{array}{rr} 1 & 3 \\ 0 & - 2 \end{array}] \cdot [\begin{array}{rrr} - 2 & 0 & 11 \\ 5 & 4 & - 9 \end{array}]$

Perform the row operations indicated.

8.

$[\begin{array}{rrr} 2 & 0 & 5 \\ - 3 & 7 & 1 \end{array}] \overset{\begin{matrix} R_{1} + R_{2} \to R_{2} \end{matrix}}{\to}$

9.

$[\begin{array}{rr} - 1 & 4 \\ 8 & 6 \\ 0 & - 2 \end{array}] \overset{\begin{matrix} 2 R_{3} + R_{1} \to R_{1} \end{matrix}}{\to}$

For problems 10 and 11, provide an example using $2 \times 2$ matrices to either illustrate or refute each of the properties below.

10.

Associative Property of Multiplication

11.

Commutative Property of Multiplication