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Pre-Calculus
Determinants and Cramer’s Rule
Main Concepts:
Determinants of Matrices
Minors and Cofactors
Determinant of an n x n Matrix
Cramer’s Rule
Determinants of Matrices
Determinant- a number associated with square matrices, written as det A or |A|
...

2 x 2 Matrix
A= [ a11 a12]
[a21 a22 ]

The determinant of this 2 x 2 matrix is written as follows: |A|= |a11 a12|
|a21 a22|

Examples of Determinants of a 2 x 2 Matrix
1
...
For this example, you would cross
multiply 6 and 13, which gives you 294
...
Subtract 468 from 294 to get -174
...

2
...
Subtract -36 from 1200
to give you a total of 1236
...

Minors and Cofactors
Any n x n matrix with a value of n being less than one is defined as follows:

The minor (Mij) of aij is the determinant of the matrix (n – 1) (n – 1) and is found by
canceling row i and column j from aij
...
Find M12 and A12 for the Matrix [15 20]
[7 -5]
To find M12 and A12 delete row 1 and column 2
[15 20]
[7 -5]
M12=7, A12 = (-1)1+2 M12 = (-1)3 (7) = -7
A12 = -7
2
...
This is referred to as “expanding by the first row”
...
|2
|-5
|1

4
12
3

1|
20|
6|

|2
|-5
|1

4
12
3

1|
1+1
1+2
20| = 2(-1)
|12 20| + 1(-1) |-5
6|
|3
6|
|1

20| + (-2)(-1)
6|

1+3

|-5 12|
|1
3|

2(12(6)-3(20) – 4(-5(6) – 1(20))- -1(-5(3) – 1(12))
2(12)-4(-50)-1(-27)=251
=251
Determinant of an n x n Matrix
The determinant of an n x n matrix is found by multiply each value in row 1 by its
cofactor then adding the sums
...
+a1nA1n
Cramer’s Rule
Cramer’s Rule is a method of solving systems of equations through determinants
...

a12x +a13y=b1
a22x +a23y=b2
Define the determinants where D is the determinant of the coefficient matrix of the
system
...
By Cramer’s Rule, the system has exactly one solution
only if

D≠0
...
By Cramer’s
Rule, the system has exactly one solution only if D≠0

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