matrices determinante
Precalculus Notes: Determinants & Cramer's Rule
1. What is a Determinant?
A determinant is a single number calculated from a square matrix.
It is NOT another matrix.
Example
[
A=
\begin{bmatrix}
3&-4\
2&5
\end{bmatrix}
]
The determinant is
[
|A|=23
]
Notice:
Matrix = table of numbers
Determinant = one number
2. Why is the Determinant Important?
The determinant tells us whether a system has one unique solution.
If
[
|A|\neq0
]
✔ One unique solution
If
[
|A|=0
]
✘ No unique solution
There may be:
infinitely many solutions
or no solution
3. Determinant of a 2 × 2 Matrix
Given
[
A=
\begin{bmatrix}
a&b\
c&d
\end{bmatrix}
]
The determinant is
[
\boxed{|A|=ad-bc}
]
Memory Trick
a b
\ /
X X
/ \
c d
Multiply the first diagonal.
Subtract.
Multiply the second diagonal.
Example 1
[
\begin{bmatrix}
3&-4\
2&5
\end{bmatrix}
]
First diagonal
[
3\times5=15
]
Second diagonal
[
2\times(-4)=-8
]
Subtract
[
15-(-8)=23
]
Answer
[
\boxed{23}
]
Example 2
[
\begin{bmatrix}
2&7\
5&1
\end{bmatrix}
]
First diagonal
[
2(1)=2
]
Second diagonal
[
5(7)=35
]
Subtract
[
2-35=-33
]
Answer
[
\boxed{-33}
]
4. Determinant Notation
Two ways
[
det(A)
]
or
[
|A|
]
Example
[
\begin{bmatrix}
3&4\
5&6
\end{bmatrix}
]
can be written as
[
|A|
]
Do not confuse this with absolute value.
If the bars contain:
a number → absolute value
a matrix → determinant
5. Determinant of a 3 × 3 Matrix
Suppose
[
A=
\begin{bmatrix}
1&2&3\
4&5&6\
7&8&9
\end{bmatrix}
]
A 3 × 3 determinant cannot use
[
ad-bc
]
Instead, use minors and cofactors.
6. Minor
A minor is the determinant of the smaller matrix left after deleting one row and one column.
Example
Find the minor of
[
a_{11}
]
Delete
Row 1
Column 1
Original
[
\begin{bmatrix}
1&2&3\
4&5&6\
7&8&9
\end{bmatrix}
]
Remove row 1 and column 1.
Remaining matrix
[
\begin{bmatrix}
5&6\
8&9
\end{bmatrix}
]
Find its determinant.
[
5(9)-6(8)
]
[
45-48=-3
]
That number is the minor.
7. Cofactor Signs
Every minor gets a sign.
Use this pattern.
[
\boxed{
\begin{matrix}
+&-&+\
-&+&-\
+&-&+
\end{matrix}
}
]
Memorize it.
8. Finding a 3 × 3 Determinant
Choose any row or column.
Most people choose the row with the most zeros.
Multiply
(entry)
×
(cofactor)
Then add everything.
Formula using the first row
[
|A|
a_{11}C_{11}
+
a_{12}C_{12}
+
a_{13}C_{13}
]
where
C = cofactor.
9. Steps for a 3 × 3 Determinant
Choose a row or column.
Delete row and column.
Find the 2×2 determinant (minor).
Apply the + − + sign.
Multiply by the original entry.
Repeat.
Add all results.
10. Cramer's Rule
Cramer's Rule solves systems using determinants.
Instead of elimination,
you
Find determinants.
Divide.
11. Step 1: Find D
Use only the coefficients.
Example
[
\begin{cases}
2x+3y=21\
x+2y=12
\end{cases}
]
Coefficient matrix
[
D=
\begin{bmatrix}
2&3\
1&2
\end{bmatrix}
]
Determinant
[
|D|
2(2)-3(1)
1
]
12. Step 2: Find (D_x)
Replace the x-column with constants.
Original
[
\begin{bmatrix}
2&3\
1&2
\end{bmatrix}
]
Replace x-column.
[
D_x=
\begin{bmatrix}
21&3\
12&2
\end{bmatrix}
]
Determinant
[
21(2)-3(12)
42-36
6
]
Now
[
x=\frac{|D_x|}{|D|}
\frac61
6
]
13. Step 3: Find (D_y)
Replace y-column.
[
D_y=
\begin{bmatrix}
2&21\
1&12
\end{bmatrix}
]
Determinant
[
2(12)-21(1)
24-21
3
]
Then
[
y=\frac{|D_y|}{|D|}
3
]
Solution
[
(x,y)
(6,3)
]
14. Cramer's Rule for Three Variables
Suppose
[
\begin{cases}
x+y+z=13\
2x-y+z=13\
x+2y-z=8
\end{cases}
]
Original coefficient matrix
[
D=
\begin{bmatrix}
1&1&1\
2&-1&1\
1&2&-1
\end{bmatrix}
]
To find
x
Replace the first column.
[
D_x=
\begin{bmatrix}
13&1&1\
13&-1&1\
8&2&-1
\end{bmatrix}
]
Then
[
x=\frac{|D_x|}{|D|}
]
y
Replace the second column.
[
y=\frac{|D_y|}{|D|}
]
z
Replace the third column.
[
z=\frac{|D_z|}{|D|}
]
15. Which Column Do I Replace?
Variable | Replace |
|---|---|
x | Column 1 |
y | Column 2 |
z | Column 3 |
Replace it with the constants.
16. Why Does Cramer's Rule Work?
The determinant tells us whether the coefficient matrix has a unique solution.
If
[
|D|=0
]
Cramer's Rule cannot be used.
If
[
|D|\neq0
]
There is one unique solution.
17. Summary
2 × 2 Determinant
[
|A|=ad-bc
]
Minor
Delete one row and one column.
Find the determinant of what remains.
Cofactor Signs
[
\begin{matrix}
+&-&+\
-&+&-\
+&-&+
\end{matrix}
]
Cramer's Rule
Find (D).
Replace one column.
Find the new determinant.
Divide.
18. Exam Tips
✓ Determinant = one number, not a matrix.
✓ Determinants only exist for square matrices.
✓ 2 × 2 determinant:
[
ad-bc
]
✓ Minor = delete one row and one column.
✓ Cofactor signs alternate:
− +
− + −
− +
✓ For Cramer's Rule:
Replace only one column at a time.
Use the constants.
Divide by the original determinant.
✓ If the original determinant is 0, Cramer's Rule does not produce a unique solution.