Algebra -- Trig Evaluate the Determinant: |

Algebra -- Trig

Evaluate the determinant: | 3-9 |

Determinant of a square matrix can be solved by the following equation: A = ad -- bc, where a = 3, b = 9, c = 6, and d = 4. Therefore, A = (3)(4) -- (9)(6) = 12 -- 54 = -42

Solve the following system of equations using matrices:

y + 4z = 6, 2x + z = 1, x + 5y + z = -9

[ 1-5-1 | -9 ]

Row 2: R2 -- 2R1 = [ 2-0-1 | 1 ] -- 2[ 1 -1-4 | 6 ] = [ 0-2 -7 | -11 ]

Row 3: R3 -- R1 = [ 1-5-1 | -9 ] -- [ 1 -1-4 | 6 ] = [ 0-6 -3 | -15 ]

New matrix:

[ 0-2 -7 | -11 ]

[ 0-6 -3 | -15 ]

Row 2: R2/2 = [ 0-1 -7/2 | -11/2 ]

Row 3: R3 -- 3R2 = [ 0-6 -3 | -15 ] -- 3[ 0-2 -7 | -11 ] = [ 0-0-18 | 18 ]

Converting back to system of equations:

x -- y + 4z = 6

y -- 7/2z = -11/2

18z = 18, where z = 1

Plugging in:

y -- 7/2(1) = -11/2 y = -11/2 + 7/2 = -4/2 = -2

x -- (-2) + 4(1) = 6 x = 6 -- 6 = 0

Answer: x = 0, y = -2, z = 1.

Write the augmented matrix for the following system of equations:

x -- 7y + z = 15, y +6z = 14, z = 11

Answer:

[ 1 -7-1 | 15 ]

[ 0-1-6 | 14 ]

[ 0-0-1 | 11 ]

4. Perform the matrix row operation(s) and write the new matrix [ 2, -4, 1| 4]

[ -5, 0, 1| -3]

[ -1, 5, -2| -1]

-3R1 + R2

-3R1 = -3[ 2 -4-1 | 4 ] = [ -6-12 -3 | -12 ]

-3R1 + R2 = [ -6-12 -3 | -12 ] + [ -5-0-1 | -3] = [ -11-12 -2 | -15 ]

Answer: [ -11-12 -2 | -15 ]

5. Use Cramer's rule to solve the given system: 2x + 4y =14, 4x + 3y =18

In matrix format, one gets:

[ 2-4 ] [ x ] = [ 14 ]

[ 4-3 ] [ y ] = [ 18 ]

Using Cramer's rule, one gets:

x = [(14)(3) -- (4)(18)] / [(2)(3) -- (4)(4)] = (42 -- 72) / (6 -- 16) = -30 / -10 = 3

y = [(2)(18) -- (14)(4)] / [(2)(3) -- (4)(4)] = (36 -- 56) / (6 -- 16) = -20 / -10 = 2

Answer: x = 3, y = 2.

6. Write the augmented matrix for the following system of equations:

2x + 7z = 64, 8y + 6z = 40, 3x -- 2y + 4z = 46

Answer:

[ 2-0-7 | 64 ]

[ 0-8-6 | 40 ]

[ 3 -2-4 | 46 ]

7. Write the augmented matrix for the following system of equations:

7x + 4y + 5z = 62, 5x + 3y + 2z = 47, 3x + 7y + 8z = 63

Answer:

[ 7-4-5 | 62 ]

[ 5-3-2 | 47 ]

[ 3-7-8 | 63 ]

8. Use Cramer's rule to solve the given system: 4x + 5y =12, 3x + y = -2

In matrix format, one gets:

[ 4-5 ] [ x ] = [ 12 ]

[ 3-1 ] [ y ] = [ -2 ]

Using Cramer's rule, one gets:

x = [(12)(1) -- (5)(-2)] / [(4)(1) -- (5)(3)] = (12 -- (-10)) / (4 -- 15) = 22 / -11 = -2

y = [(4)(-2) -- (12)(3)] / [(4)(1) -- (5)(3)] = (-8 -- 36) / (4 -- 15) = -44 / -11 = 4

Answer: x = -2, y = 4.

9. Evaluate the determinant: | -9, -7|

| 4, 5|

Determinant of a square matrix can be solved by the following equation: A = ad -- bc, where a = -9, b = -7, c = 4, and d = 5. Therefore, A = (-9)(5) -- (-7)(4) = -45 -- (-28) = -17

Answer: -17.

10. Using x, y, z, and w for the variables, write the system of linear equations represented by the given augmented matrix:

[ 9, 1, 0, 8| 6]

[-1, 5, 1, 0|-12]

[7, 0, 0, 3,| -9]

[0, 4, 0,-4| 8]

Answer:

9x + y + 8w = 6

-x + 5y + z = -12

7x + 3w = -9

4y -- 4w = 8

11. Use Cramer's rule to solve the given system: 2x + 2y = 32, 2x -- 3y = -3

In matrix format, one gets:

[ 2-2 ] [ x ] = [ 32 ]

[ 2 -3 ] [ y ] = [ -3 ]

Using Cramer's rule, one gets:

x = [(32)(-3) -- (2)(-3)] / [(2)(-3) -- (2)(2)] = (-96 -- (-6)) / (-6 -- 4) = -90 / -10 = 9

y = [(2)(-3) -- (32)(2)] / [(2)(-3) -- (2)(2)] = (-6 -- 64) / -6 -- 4) = -70 / -10 = 7

Answer: x = 9, y = 7.

12. Write the system of linear equations represented by the following augmented matrix:

[ 9, 4, 5| -2]

[ 5, 0, 9| 4]

[ 8, 5, 0| 2]

Answer:

9x + 4y + 5z = -2

5x + 9z = 4

8x + 5y = 2

13. Evaluate the determinant: | 2-3|

| 3-2|

Determinant of a square matrix can be solved by the following equation: A = ad -- bc, where a = 2, b = 3, c = 3, and d = 2. Therefore, A = (2)(2) -- (3)(3) = 4 -- 9 = -5