A mathematical theorem states that a matrix is singular if and only if its determinant is zero. Is the following matrix singular?

### Watch This

Watch the first part of this video, until about 4:00.

James Sousa: Evaluating Determinants of a 2x2 and 3x3 Matrix

### Guidance

Each square matrix has a real number value associated with it called its
**
determinant.
**
This value is denoted by
or
.

Finding the
**
Determinant of a
matrix:
**

Finding the
**
Determinant of a
matrix:
**
To begin, we will repeat the first two columns after matrix. Next, calculate the products and sums as shown below and find the difference. The result is the determinant of the
matrix.

Using the determinant to find the
**
Area of a Triangle
**
in the coordinate plane:

We can find the area of a triangle with vertices and using the formula below

where the accounts for the possibility that the determinant could be negative but area should always be positive.

Using the
**
calculator to find the determinant of a matrix:
**
If you are using a TI-83 or TI-84, access the Matrix menu by either pressing MATRIX or (
MATRIX). Now you can choose to EDIT matrix
. Change the dimensions as needed and enter the data values. Now return to the home screen (
MODE QUIT) and return to the MATRIX menu. Arrow over to MATH and select 1:det( by pressing ENTER. Go into the MATRIX menu once more to select 1:
under the NAMES column. Press ENTER. Your screen should show
at this time. Press ENTER once more and the result will be your determinant. These directions work for square matrices of any size.

#### Example A

Find the

**
Solution:
**
Using the rule above for a
matrix, the determinant can be found as shown:

#### Example B

Find the

**
Solution:
**

First, we need to repeat the first two columns. Then we can find the diagonal products as shown:

#### Example C

Find the area of the triangle with vertices (2, -1), (4, 5) and (8, 1)

**
Solution:
**

The first step is to set up our matrix and find the determinant as shown:

Now we can multiply this determinant, -32, by (we will multiply by the negative in order to have a positive result) to get 16. So the area of the triangle is 16 .

**
Intro Problem Revisit
**
To find the determinant, we first need to repeat the first two columns. Then we can find the diagonal products as shown:

The determinant is not zero and therefore the matrix is not singular.

### Guided Practice

Find the determinants of the matrices below.

1.

2.

3. Find the area of the triangle with vertices (-5, 2), (8, -1) and (3, 9)

#### Answers

1.

2.

3.

So the area is .

### Practice

Find the determinants of the matrices. Use your calculator to check your answers.

- .

- .

- .

- .

- .

- .

- .

- .

- .

Find the area of each triangle with vertices given below.

- (2, -1), (-5, 2) and (0, 6)
- (-8, 12), (10, 5) and (1, -4)
- (-7, 2), (8, 0) and (3, -4)

Find the value of in the matrices below.

- .

- .

- .