1. Warmup:
Find the product, if possible.
−6 4 − 

2. 3-7 Solving systems of equations using Cramer’s Rule
Evaluate determinants, find area of a triangle, and solve systems.

3. Every square matrix has a determinant
Every square matrix has a determinant. The determinant of a 2 × 2 matrix can be evaluated by calculating the difference of the products of the two diagonals. The determinant of a 3 × 3 matrix can be evaluated using the diagonal rule.

4. 26. Evaluate each determinant. −7 12 5 6

6. Evaluate the determinant.
2 0 −6 −3 −4 −5 −2 5 8

7. 49. Archaeologists found whale bones at coordinates (0, 3), (4, 7), and (5, 9). If the units of the coordinates are meters, find the area of the triangle formed by these finds.

8. Cramer's Rule can be used to solve systems of equations
Cramer's Rule can be used to solve systems of equations. To use Cramer's Rule to solve a system of equations,
write each equation in standard form,
then set each variable equal to a fraction whose denominator and numerator are both determinants.
A system has no solution if the value of the determinant in the denominator is zero.

9. Use Cramer’s Rule to solve the system of equations.

10. Cramer's Rule One advantage of using Cramer's Rule to solve a system of three linear equations in three variables is that if the system has a unique solution, Cramer's Rule can give the value of any one of the variables without having to solve for the others. If the determinant is zero, then Cramer's Rule does not give the solution, but it does indicate that the system is either dependent or inconsistent.

11. Use Cramer’s Rule to solve the system of equations
8x – 4y + 7z = 34
5x + 6y + 3z= –21
3x + 7y – 8z = –85