# Determinants and Cramer's Rule - Video Tutorials & Practice Problems

## Determinants of 2Ã—2 Matrices

Evaluate the determinant of the matrix**.**

ï»¿$\frac{107}{3}$ï»¿

ï»¿$\frac{227}{6}$ï»¿

33

12

## Cramer's Rule - 2 Equations with 2 Unknowns

Write each equation in standard form and use Cramer's Rule to solve the system**.**

ï»¿$y=-3x+4$ï»¿

ï»¿$-2x=7y-9$ï»¿

$x=1,y=1$

$x=\xe2\u02c6\u20191,y=1$

$x=1,y=\xe2\u02c6\u20191$

$x=\xe2\u02c6\u20191,y=\xe2\u02c6\u20191$

Write each equation in standard form and use Cramer's Rule to solve the system**.**

ï»¿$y-9x=-3$ï»¿

ï»¿$-3x=4y-1$ï»¿

ï»¿$y=3,x=0$ï»¿

ï»¿$x=0,y=3$ï»¿

ï»¿$x=-\frac13,y=1$ï»¿

ï»¿$x=\frac13,y=0$ï»¿

## Determinants of 3Ã—3 Matrices

Evaluate the determinant of the matrix**.**

**ï»¿**

165

9

63

25

## Cramer's Rule - 3 Equations w/ 3 Unknowns

Solve the system of equations using Cramer's Rule**.**

ï»¿$4x+2y+3z=6$ï»¿

ï»¿$x+y+z=3$ï»¿

ï»¿$5x+y+2z=5$ï»¿

$x=\xe2\u02c6\u20192,y=\xe2\u02c6\u20198,z=4$

$x=1,y=4,z=\xe2\u02c6\u20192$

$x=2,y=8,z=\xe2\u02c6\u20194$

$x=\xe2\u02c6\u20191,y=\xe2\u02c6\u20194,z=2$

## Do you want more practice?

- What is the value of ?
- Evaluate each determinant in Exercises 1â€“10. 5 7 2 3
- What expression in x represents ?
- Evaluate each determinant in Exercises 1â€“10. - 4 1 5 6
- Evaluate each determinant in Exercises 1â€“10. - 7 14 2 - 4
- What is the value of x if = 9?
- Evaluate each determinant. See Example 1.
- Evaluate each determinant in Exercises 1â€“10. - 5 - 1 - 2 - 7
- Evaluate each determinant in Exercises 1â€“10. - 5 - 1 - 2 - 7
- Evaluate each determinant. See Example 1.
- Evaluate each determinant in Exercises 1â€“10. 1/2 1/2 1/8 - 3/4
- For Exercises 11â€“22, use Cramer's Rule to solve each system. x + y = 7 x - y = 3
- Evaluate each determinant. See Example 1.
- Evaluate each determinant. See Example 1.
- For Exercises 11â€“22, use Cramer's Rule to solve each system. 12x + 3y = 15 2x - 3y = 13
- For Exercises 11â€“22, use Cramer's Rule to solve each system. 12x + 3y = 15 2x - 3y = 13
- Evaluate each determinant. See Example 1.
- For Exercises 11â€“22, use Cramer's Rule to solve each system. 4x - 5y = 17 2x + 3y = 3
- For Exercises 11â€“22, use Cramer's Rule to solve each system. x + 2y = 3 3x - 4y = 4
- Find the cofactor of each element in the second row of each matrix. See Example 2.
- Find the cofactor of each element in the second row of each matrix. See Example 2.
- For Exercises 11â€“22, use Cramer's Rule to solve each system. 3x - 4y = 4 2x + 2y = 12
- For Exercises 11â€“22, use Cramer's Rule to solve each system. 3x - 4y = 4 2x + 2y = 12
- Evaluate each determinant. See Example 3.
- For Exercises 11â€“22, use Cramer's Rule to solve each system. 2x = 3y + 2 5x = 51 - 4y
- In Exercises 23â€“30, use expansion by minors to evaluate each determinant. 3 0 0 2 1 - 5 2 5 - 1
- Evaluate each determinant. See Example 3.
- Evaluate each determinant. See Example 3.
- In Exercises 23â€“30, use expansion by minors to evaluate each determinant. 3 1 0 - 3 4 0 - 1 3 - 5
- In Exercises 23â€“30, use expansion by minors to evaluate each determinant. 3 1 0 - 3 4 0 - 1 3 - 5
- Evaluate each determinant. See Example 3.
- In Exercises 23â€“30, use expansion by minors to evaluate each determinant. 1 1 1 2 2 2 - 3 4 - 5
- In Exercises 23â€“30, use expansion by minors to evaluate each determinant. 0.5 7 5 0.5 3 9 0.5 1 3
- In Exercises 31â€“36, use the alternative method for evaluating third-order determinants on here to evaluate eac...
- In Exercises 31â€“36, use the alternative method for evaluating third-order determinants on here to evaluate eac...
- In Exercises 31â€“36, use the alternative method for evaluating third-order determinants on here to evaluate eac...
- Evaluate each determinant.
- In Exercises 37â€“44, use Cramer's Rule to solve each system. x + y + z = 0 2x - y + z = - 1 - x + 3y - z = - ...
- In Exercises 37â€“44, use Cramer's Rule to solve each system. x + y + z = 0 2x - y + z = - 1 - x + 3y - z = - ...
- Evaluate each determinant.
- In Exercises 37â€“44, use Cramer's Rule to solve each system. 4x - 5y - 6z = - 1 x - 2y - 5z = - 12 2x - y = 7
- Evaluate each determinant.
- In Exercises 37â€“44, use Cramer's Rule to solve each system. x + y + z = 4 x - 2y + z = 7 x + 3y + 2z = 4
- In Exercises 37â€“44, use Cramer's Rule to solve each system. x + 2z = 10 2y - z = - 5 2x + 3y = 13
- In Exercises 37â€“44, use Cramer's Rule to solve each system. x + 2z = 10 2y - z = - 5 2x + 3y = 13
- In Exercises 45â€“48, explain why the system of equations cannot be solved using Cramer's Rule. Then use Gaussia...
- In Exercises 46â€“51, evaluate each determinant.
- In Exercises 45â€“48, explain why the system of equations cannot be solved using Cramer's Rule. Then use Gaussia...
- Use the determinant theorems to evaluate each determinant. See Example 4.
- Solve each equation. = 8
- Solve each equation. = 2x
- Evaluate each determinant in Exercises 49â€“52. 4 2 8 - 7 - 2 0 4 1 5 0 0 5 4 0 0 - 1
- In Exercises 46â€“51, evaluate each determinant.
- Evaluate each determinant in Exercises 49â€“52. 4 2 8 - 7 - 2 0 4 1 5 0 0 5 4 0 0 - 1
- Use the determinant theorems to evaluate each determinant. See Example 4.
- Evaluate each determinant in Exercises 49â€“52. - 2 - 3 3 5 1 - 4 0 0 1 2 2 - 3 2 0 1 1
- In Exercises 46â€“51, evaluate each determinant.
- Use the determinant theorems to evaluate each determinant. See Example 4.
- In Exercises 46â€“51, evaluate each determinant.
- In Exercises 53â€“54, evaluate each determinant. | | 3 1| |7 0| | | |- 2 3| |1 5| | | | | | 3 0| |9 - 6| | | |...
- In Exercises 52â€“55, use Cramer's Rule to solve each system.
- Use the determinant theorems to evaluate each determinant. See Example 4.
- In Exercises 52â€“55, use Cramer's Rule to solve each system.
- Use the determinant theorems to evaluate each determinant. See Example 4.
- In Exercises 55â€“56, write the system of linear equations for which Cramer's Rule yields the given determinants...
- In Exercises 57â€“60, solve each equation for x. |- 2 x| | | = 32 | 4 6|
- Use the determinant theorems to evaluate each determinant. See Example 4.
- In Exercises 57â€“60, solve each equation for x. |1 x - 2| |3 1 1| = - 8 |0 - 2 2|
- Use the determinant theorems to evaluate each determinant. See Example 4.
- In Exercises 57â€“60, solve each equation for x. |1 x - 2| |3 1 1| = - 8 |0 - 2 2|
- Use the determinant theorems to evaluate each determinant. See Example 4.
- Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the soluti...
- Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the soluti...
- Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the soluti...