3.7 Evaluate Determininants and Apply Cramer's Rule

Determinant

A real number associated with any square matrix A and denoted by det A or |a|

Cramer's Rule

A method to solve a system of linear equations using the determinants of matrices

Coefficient matrix

The coefficient matrix of the linear

ax + by = e

system is

cx + dy = f

THE DETERMINANT OF A MATRIX

Determinant of a 2 x 2 matrix

det = = _ad_ - _cd_

Determinant of a 3 ´ 3 matrix

det =

= (aei + bfg + cdh ) - (gec + hfa + idb)

Your Notes

Example 1

Evaluate determinants

Evaluate the determinant of the matrix.

a. b.

Solution

a. = 4 (__2__) - 5(__-3__) = _8 - ( - 15)_ = __23__

b. = _( 18 + 0 - 4)_ - _(0 + 8 - 6)_

AREA OF A TRIANGLE

The area of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is given by

Area = ±_{____}

where the symbol ± indicates that the appropriate sign should be chosen to yield a _positive_ value.

Example 2

The area of a triangle

Find the area of the triangle shown.

Area = ± _{______}

= ± [ (-3 - 9 + 2) - (3 + 6 + 3) ] = _11_

Complete the following exercises.

1. Evaluate the determine of

26

2. Find the area of a triangle with vertices (1, 4), (3,1), and (-1, 0).

7

CRAMER'S RULE FOR A 2 ´ 2 SYSTEM

Let A be the coefficient matrix of this linear system:

ax + by = e

ex + dy = f

If det A ¹ _0_, then the system has _exactly one solution_

_{x = _______and y = ________}

det A det A

Example 3

Use Cramer's rule fora2´2system

Use Cramer’s Rule to solve this system

3x + 2y = -4

2x - 7y = -11

Evaluate the determinant of the _coefficient_ matrix.

= _- 21 - 4_ = _-25_

Apply Cramer's rule because the determinant is not __0__.

_{x =_____________= _28 - (-22)_= __-2_}

_-25_ ___-2__

_{y = ____________ = _-33 - (-8)_ = __1_}

__-25_ ___-25_

The solution is (_-2_, _1_ ).

Remember to check your solution in each of the original equations

CRAMER'S RULE FOR A 3 x 3 SYSTEM

Let A be the coefficient matrix of the linear system:

ax + by + cz = j

dx + ey + fz = k

gx + hy + iz = l

If det A ¹ _0_, then the system has _exactly one_ solution.

_{x = ____________,y = __________,z = _________}

det A det A det A

Example 4

Use Cramer's rule for a 3 x 3 system

Use Cramer's rule to solve this system:

3x + 4y - z = 9

-2x - 3y + 4z = -14

4x - y + z = -18

Evaluate the determinant of the _coefficient_ matrix.

= _(-9 + 64 - 2) - (12 - 12 - 8)_

=__61__

Apply Cramer's rule because the determinant is not _0_.

x =

y =

z =

Your Notes

Checkpoint Use Cramer's rule to solve the system.

3. 3x + y = 4

5x + 4y = - 5

(3,-5)

4. 2x - 3y -2z = -10

-x + 2y + 3z = 14

4x + y + 2z = -4

(-3,-2,5)

Homework

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