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
a b 
The determinant of the matrix 
 is
c d 
a b
 ad  bc .
c d
“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:
a.
2 2
 26  3 2  12  6  18
3 6
4 2
 43  6 2  12  12  0
6
3
“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.”
1 5
 1 4  65  34
c.
6 4
b.
Finding Determinant of 3x3 Matrix ~ Expansion by Minors
a b c
e f
d f
d e
d e f a
b
c
 aej  ahf  bdj  bgf  cdh  cge
h j
g j
g h
g h j
(This is the expansion by the first row.)
To set-up the minor matrix, ignore the row and column that contains the coefficient.
a b c
a b c
a b c
a b c
d e f  ad e f bd e f c d e f
g h j
g h j
g h j
g h j
Ex. 2: Find the determinant using expansion by minors:
2 1 3
1 4
1 4
1 1
1 1 4  2
1
3
6 2
3 2
3 6
3 6 2
 22  24   12   12  36   3 
 2 22  114   39 
 44  14  27
 30  27
 57
Finding Determinant of 3x3 Matrix ~ Diagonals Method
a b c a b
d e f d e  aej  bfg  cdh  gec  hfa  jdb
g h j g h
“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.
2 1 3 -2 1
1 1 4 1 1  2 * 1 * 2  1 * 4 *  3   3 * 1 * 6   3  * 1 * 3   6 * 4 *  2  2 * 1 * 1
3 6 2 -3 6
 4  12  18  9  48  2
 16  27  46
 57
Applications of Determinants (besides to find inverses)
Ex. 4 Find the area of a triangle whose vertices are at 0,0 , 10,25 and 28,20 .
“Formula”: Let x1,y 1  , x 2 ,y 2  ,and x 3 ,y 3  be the vertices of a triangle. Then
x1 y 1 1
1
A   x2 y 2 1
2
x3 y 3 1
(Why 1’s in the 3rd column? Multiplying by 1 does not change a number)
0
0
1 0
0
1
A   10  25 1 10  25
2
28  20 1 28  20
1
0  0  200   700  0  0
2
1
  500   250 square units
2
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  4,1 and 3,5 .
x
“Formula”: x1
x2
y
1
y1 1  0
y2 1
x
y 1 x
y
4 1 1 4 1 0
3 5 1 3 5
x  3 y  20  3   5 x    4 y   0
x  3 y  17  5 x  4 y  0
6 x  7 y  17
Check using point-slope form of a line:
1  5
6
m

43
7
6
6
17
y  1   x  4   y   x 
 6 x  7 y  17
7
7
7
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 __________________________.
Finding the Determinant of a 2 x 2 Matrix
a b 
The determinant of the matrix 
 is
c d 
.
Example 1: Find the determinant of the following matrices:
a.
2 2
3 6
b.
4 2
6
3
c.
1 5
6 4
Finding Determinant of 3x3 Matrix ~ Expansion by Minors
a b c
e f
d f
d e
d e f a
b
c

h j
g j
g h
g h j
(This is the expansion by the first row.)
To set-up the minor matrix, ignore the row and column that contains the coefficient.
a b c
a b c
a b c
a b c
d e f  ad e f bd e f c d e f
g h j
g h j
g h j
g h j
Ex. 2: Find the determinant using expansion by minors:
2 1 3
1 1 4
3 6 2
Finding Determinant of 3x3 Matrix ~ Diagonals Method
a b c a b
d e f d e
g h j g h
Ex. 3 Find the determinant for the matrix in example 2 using the diagonals method.
2 1 3 -2 1
1 1 4 1 1
3 6 2 -3 6
Applications of Determinants (besides to find inverses)
Ex. 4 Find the area of a triangle whose vertices are at 0,0 , 10,25 and 28,20 .
Ex. 5 Find the equation of a line that contains  4,1 and 3,5 .
Check using point-slope form of a line: