1. § 3.5
Determinants and Cramer’s Rule

2. Determinants Evaluate the determinant of the matrix. EXAMPLE SOLUTION
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 3.5

3. Determinants The determinant of the matrix is defined by
Definition of the Determinant of a 2 by 2 matrix
The determinant of the matrix
is defined by
The determinant of a matrix may be positive or negative.
The determinant can also have 0 as its value.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 3.5

4. Solving a Linear System in Two Variables Using Determinants
Determinants – Cramer’s Rule
Solving a Linear System in Two Variables Using Determinants
Cramer’s Rule If
then
and
where
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 3.5

5. Determinants – Cramer’s Rule
EXAMPLE
Use Cramer’s rule to solve the system:
SOLUTION
Because we’re using Cramer’s rule to solve this system, we must first write the system in standard form.
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 3.5

6. Determinants – Cramer’s Rule
CONTINUED
Now we need to determine the different values as defined in Cramer’s rule:
Therefore
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 3.5

7. Determinants – Cramer’s Rule
CONTINUED
Now we can determine x and y.
Therefore, the solution to the system is (-2,-1). As always, the ordered pair should be checked by substituting these values into the original equations.
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 3.5

8. Evaluating the Determinant of a 3x3 Matrix
Determinants
Evaluating the Determinant of a 3x3 Matrix
1) Each of the three terms in the definition contains two factors – a numerical factor and a second-order determinant.
2) The numerical factor in each term is an element from the first column of the third-order determinant.
3) The minus sign precedes the second term.
4) The second-order determinant that appears in each term is obtained by crossing out the row and the column containing the numerical factor.
The minor of an element is the determinant that remains after deleting the row and column of that element. For this reason, we call this method expansion by minors.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 3.5

9. Determinants Evaluate the determinant of the matrix.
EXAMPLE
Evaluate the determinant of the matrix.
SOLUTION
The minor for 2 is
The minor for -2 is
The minor for 4 is
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 3.5

10. Determinants Therefore
CONTINUED
Therefore
= 2[(3)(8) – (-6)(5)] + 2[(1)(8) – (-6)(0)] + 4[(1)(5) – (3)(0)]
= 2[24 – (-30)] + 2[8 – 0] + 4[5 – 0]
= 2[ ] + 2[8 – 0] + 4[5 – 0]
= 2[54] + 2[8] + 4[5] = = 144
Therefore, the determinant of the matrix is 144.
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 3.5

11. Determinants – Cramer’s Rule
EXAMPLE
Use Cramer’s rule to solve the system:
SOLUTION
We first need to determine the different values as defined in Cramer’s rule:
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 3.5

12. Determinants – Cramer’s Rule
CONTINUED
Therefore
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 3.5

13. Determinants – Cramer’s Rule
CONTINUED
Now I can determine x, y and z.
Therefore, the solution to the system is (-2, 3/5, 12/5). As always, the ordered pair should be checked by substituting these values into the original equations.
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 3.5