Lesson 8 The Determinant of a Matrix Solidify Understanding

Learning Focus

Examine a new strategy for finding the multiplicative inverse of a matrix.

Is it possible to divide by a matrix? How is division defined?

How does the idea that “dividing by $0$ is undefined” or that “ $0$ doesn’t have a multiplicative inverse” show up in matrix multiplication?

Open Up the Math: Launch, Explore, Discuss

In the previous task we learned how to find the multiplicative inverse of a matrix. Use that process to find the multiplicative inverse of the following two matrices.

1.

$[\begin{matrix} 5 & 1 \\ 6 & 2 \end{matrix}]$

2.

$[\begin{matrix} 6 & 2 \\ 3 & 1 \end{matrix}]$

3.

Were you able to find the multiplicative inverse for both matrices?

There is a number associated with every square matrix called the determinant of the matrix. If the determinant is not equal to $0$ , then the matrix has a multiplicative inverse.

For a $2 \times 2$ matrix the determinant can be found using the following rule: (note: the vertical lines, rather than the square brackets, which are used to indicate that we are finding the determinant of the matrix).

The determinant of a $2 \times 2$ matrix is calculated by $| \begin{matrix} a & b \\ c & d \end{matrix} | = a d - b c$ .

4.

Using this rule, find the determinant of the two matrices given in problems 1 and 2.

The absolute value of the determinant of a $2 \times 2$ matrix, $[\begin{matrix} a & b \\ c & d \end{matrix}]$ , can be visualized as the area of a parallelogram, constructed as follows:

Draw one side of the parallelogram with endpoints at $(0, 0)$ and $(a, c)$ .
Draw a second side of the parallelogram with endpoints at $(0, 0)$ and $(b, d)$ .
Locate the fourth vertex that completes the parallelogram.

(Note that the elements in the columns of the matrix are used to define the endpoints of the vectors that form two sides of the parallelogram.)

5.

Use the diagram to show that the area of the parallelogram is given by $a d - b c$ .

6.

Draw the parallelograms whose areas represent the determinants of the two matrices listed in problems 1 and 2. How does a $0$ determinant show up in these diagrams?

7.

Create a matrix for which the determinant will be negative. Draw the parallelogram associated with the determinant of your matrix on graph paper and find the area of the parallelogram. What observations can you make?

Pause and Reflect

The determinant can be used to provide an alternative method for finding the inverse of $2 \times 2$ matrix.

8.

Use the process you used previously to find the inverse of a generic $2 \times 2$ matrix whose elements are given by the variables $a$ , $b$ , $c$ , and $d$ . For now, we will refer to the elements of the inverse matrix as $M_{1}$ , $M_{2}$ , $M_{3}$ , and $M_{4}$ as illustrated in the following matrix equation. Find expressions for $M_{1}$ , $M_{2}$ , $M_{3}$ , and $M_{4}$ in terms of the elements of the first matrix, $a$ , $b$ , $c$ , and $d$ .

$[\begin{matrix} a & b \\ c & d \end{matrix}] \cdot [\begin{matrix} M_{1} & M_{2} \\ M_{3} & M_{4} \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

a.

$M_{1} =$

b.

$M_{2} =$

c.

$M_{3} =$

d.

$M_{4} =$

e.

Use your previous work to explain this strategy for finding the inverse of a $2 \times 2$ matrix: (note: the $- 1$ superscript is used to indicate that we are finding the multiplicative inverse of the matrix)

${[\begin{matrix} a & b \\ c & d \end{matrix}]}^{- 1} = \frac{1}{a d - b c} [\begin{matrix} d & - b \\ - c & a \end{matrix}]$ where $a d - b c$ is the determinant of the matrix.

Ready for More?

When working with real numbers, we know that dividing by a number is equivalent to multiplying by the reciprocal, or multiplicative inverse of that number. For example, $4 \div 2 = 4 \times \frac{1}{2} = 2$ . Of course, we can’t divide by $0$ because $0$ doesn’t have a multiplicative inverse.

How do we divide matrices? Typically, matrix division isn’t treated as an operation. Rather, we can interpret matrix division as multiplying by the multiplicative inverse. How might we find the matrix that is the “quotient “of two matrices, such as

$[\begin{matrix} 6 & 2 \\ 2 & 1 \end{matrix}] \div [\begin{matrix} 4 & 3 \\ 2 & 2 \end{matrix}]$ ?

Takeaways

The multiplicative inverse of a $2 \times 2$ matrix, $[\begin{matrix} a & b \\ c & d \end{matrix}]$ , can be found using the formula: .

When the determinant is $a d - b c = 0$ , .

Adding Notation, Vocabulary, and Conventions

The determinant of a matrix can be associated with

This number is important because

A square matrix that has a nonzero determinant

A square matrix for which the determinant is $0$

For a $2 \times 2$ matrix, $[\begin{matrix} a & b \\ c & d \end{matrix}]$ , the notation for finding the determinant of the matrix is:

For a $2 \times 2$ matrix, $[\begin{matrix} a & b \\ c & d \end{matrix}]$ , the determinant can be calculated by:

Vocabulary

determinant of a matrix
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned a second method for finding the multiplicative inverse of a $2 \times 2$ matrix. We also found ways to determine if a $2 \times 2$ matrix has a multiplicative inverse or not.

Retrieval

1.

Describe the row operations that occurred between the first and second matrix.

$[\begin{array}{rrr} 5 & 1 & 9 \\ 10 & - 7 & - 18 \end{array}] \to [\begin{array}{rrr} 5 & 1 & 9 \\ 0 & 9 & 36 \end{array}]$

2.

These matrices come from a system of linear equations. Continue using row operations to find the solutions for the system.

$[\begin{array}{rrr} 5 & 1 & 9 \\ 10 & - 7 & - 18 \end{array}] \to [\begin{array}{rrr} 5 & 1 & 9 \\ 0 & 9 & 36 \end{array}]$

3.

Determine whether the system of equations contains lines that are parallel, perpendicular, or neither.

${\begin{cases} 2 x + 4 y = 8 \\ 3 x + 6 y = 3 \end{cases}$

A.

parallel

B.

perpendicular

C.

neither