Determinant of a Matrix Teacher Notes

Determinant of a Matrix ~ Teacher Notes

Student Notes at the end

Students may find it helpful to have a colored pencil or two helpful here.

Recall: A square matrix has the same number of rows and columns.

A real number associated with each square matrix is the determinant.

Finding the Determinant of a 2 x 2 Matrix

The determinant of the matrix is

“Notice that the matrix is now surrounded by straight bars instead of brackets. The straight bars indicate determinant. So when dealing with matrices and you come across straight bars around the matrix, DO NOT think absolute value. Think DETERMINANT.”

Example 1: Find the following:

“Can you identify a relationship between the rows/columns of this matrix? When the determinant is zero, there is a column/row that is a multiple of another column/row.”

Finding Determinant of 3x3 Matrix ~ Expansion by Minors

(This is the expansion by the first row.)

To set-up the minor matrix, ignore the row and column that contains the coefficient.

Ex. 2: Find the determinant using expansion by minors:

Finding Determinant of 3x3 Matrix ~ Diagonals Method

“The first step is to rewrite the first two columns of the matrix to the left of it.

Next, we will be multiplying the entries along three down diagonals (the red arrows) and up three other diagonals (blue).

The main thing to remember is when to start out initially as the product being positive or negative. When going downward, the product is initially positive. When going upward, the product is initially negative. You can relate this to skiing: going downhill is positive since it is more fun; going upward is negative since it is generally harder.”

Ex. 3 Find the determinant for the matrix in example 2 using the diagonals method.

Applications of Determinants (besides to find inverses)

Ex. 4 Find the area of a triangle whose vertices are at ,and.

“Formula”: Let ,,and be the vertices of a triangle. Then

(Why 1’s in the 3rd column? Multiplying by 1 does not change a number)

Multiply the 500 by a positive ½ or you would get a negative area which is not possible.

“What if you were asked to find the area of a polygon with more than 3 sides? Draw diagonals from 1 vertex to split up the polygon into non-overlapping triangles. Use the above technique to find the area of each triangle and then add the individual areas together to find the total area.”

Ex. 5 Find the equation of a line that contains and.

“Formula”:

Check using point-slope form of a line:

Determinant of a Matrix ~ Student Notes

Recall: A square matrix has the same number of ______.

A real number associated with each square matrix is the ______.