3.7: Evaluate Determinants and Apply Cramer’s Rule
Determinant: Is the difference in the diagonals
Notation: Determinant of matrix A is |A|
Can only occur with a square matrix: 2x2, 3x3, 4x4, etc.
Determinant of a 2x2
a
c
b
d
ad – cb
Make up 3 examples to show them
For a 3 x 3 repeat rows to be able to complete diagonals
a
b
d
e
g
h
c
f
i
Notice the only real diagonals are aei and gec, but this leaves out several letters.
So we repeat two front columns on the outside
a
b
c
a
b
d
e
f
d
e
g
h
i
g
h
Notice the new columns are exactly like the first two in the original, Now we can find all
diagonals
Determinant is the difference in the sum of the products of the diagonals going
from top left to bottom right and the sum of the products of the diagonals going from
bottom left to top right
(aei + bfg + cdh) – (gec + hfa + idb)
Make up 3 examples to show them
Cramer’s Rule for a 2x2 System: Linear Systems can be solved by using matrices.
ax + by = e
cx + dy = f
As long as the determinant doesn’t = 0, the following is used
Steps: Make two matrices
1. Put the coefficients in a matrix
a b
c d
2. Find the determinant of the coefficient matrix ad - cb
3. for x, don’t include coefficients for x, make the following matrix
e b
f d
4. Find the determinant of the above matrix. Eb - fb
5. The value of x = x’s matrix determinant/coefficient matrix determinant
6. Follow same suite For y, don’t include coefficients for y
a e
c f
Find determinant: af – ce
Value of y = y’s matrix determinant/coeffiecient matrix determinant
Use these matrices like a formula and plug values in accordingly.
Go through example 3: I do it with them
9x + 4y = -6
3x – 5y = -21
Example they do with me step by step
6x – 2y = -16
-3x + 5y = 16
Example for them on their own
3x – 4y = -15
2x + 5y = 13
For a 3x3 x, y, and z
ax + by + cz = j
dx + ey + fz = k
gx + hy + iz = l
Steps
1. Make coefficient matrix and find the determinant.
2. Matrix for x =
j
b
c
k
e
f
l
h
i
3. Find the determinant for above matrix
X = x’s matrix determinant/coefficient matrix determinant
4. Matrix for y=
a
d
g
j
k
l
5. Find the determinant for above matrix
Y = y’s matrix determinant/coefficient matrix determinant
Use Guided Practice #7 for an example and #34
Go over example 4
Homework: 1, 2-20ev, 21, 30 – 34 evens, 42,43
c
f
i