Lesson 4 The Determinant of a Matrix Solidify Understanding

Ready

Given the system of equations ${\begin{cases} 5 x - 3 y = 3 \\ 2 x + y = 10 \end{cases}$ .

1.

Zac started solving this problem by writing $[\begin{array}{rrr} 5 & - 3 & 3 \\ 2 & 1 & 10 \end{array}] \to [\begin{array}{rrr} 1 & - 5 & - 17 \\ 2 & 1 & 10 \end{array}]$ .

Describe what Zac did to get from the matrix on the left to the matrix on the right.

2.

Lea started solving this problem by writing $[\begin{array}{rrr} 5 & - 3 & 3 \\ 2 & 1 & 10 \end{array}] \to [\begin{array}{rrr} 5 & - 3 & 3 \\ 1 & \frac{1}{2} & 5 \end{array}]$ .

Describe what Lea did to get from the matrix on the left to the matrix on the right.

3.

Using either Zac’s or Lea’s first step, continue solving the system using row reduction. Show each matrix along with notation indicating how you got from one matrix to another. Be sure to check your solution.

Set

4.

Use the determinant of each $2 \times 2$ matrix to decide which matrices have multiplicative inverses, and which do not.

a.

$[\begin{array}{rr} 8 & - 2 \\ 4 & 1 \end{array}]$

b.

$[\begin{array}{rr} 3 & 6 \\ 2 & 4 \end{array}]$

c.

$[\begin{array}{rr} 4 & 2 \\ 3 & 1 \end{array}]$

5.

Find the multiplicative inverse of each of the matrices in problem 4, provided the inverse matrix exists.

a.

$[\begin{array}{rr} 8 & - 2 \\ 4 & 1 \end{array}]$

b.

$[\begin{array}{rr} 3 & 6 \\ 2 & 4 \end{array}]$

c.

$[\begin{array}{rr} 4 & 2 \\ 3 & 1 \end{array}]$

6.

Generally, matrix multiplication is not commutative. That is, if $A$ and $B$ are matrices, typically $A \cdot B \neq B \cdot A$ . However, multiplication of inverse matrices is commutative. Test this out by showing that the pairs of inverse matrices you found in question 5 give the same result when multiplied in either order.

Go

Determine if the following pairs of lines are parallel, perpendicular, or neither. Explain how you arrived at your answer.

7.

$3 x + 2 y = 7$ and $6 x + 4 y = 9$

8.

$y = \frac{2}{3} x - 5$ and $y = - \frac{2}{3} x + 7$

9.

$y = \frac{3}{4} x - 2$ and $4 x + 3 y = 3$

10.

Write the equation of a line that is parallel to $y = \frac{4}{5} x - 2$ and has a $y$ -intercept at $(0, 4)$ .