6.6 KiB
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3 by 3 Determinants |
3 by 3 Determinants
Method 1
Consider the following matrix, which we will call A:
a | b | c |
d | e | f |
g | h | i |
Then the determinant of this matrix, denoted det(A), is given by:
det(A) = a * (e * i - h * f) - b * (d * i - f * g) + c * (d * h - e * g)
Please keep in mind the order of operations in the expression above.
For example, consider the following matrix, which we will call B:
1 | 2 | 3 |
0 | -3 | 5 |
-10 | 4 | 7 |
det(B) is given by the formula above. We apply the formula below:
det(B) = 1 * ( (-3) * 7 - 5 * 4) - 2 * ( 0 * 7 - 5 * (-10)) + 3 * (0 * 4 - (-3) * (-10)), which we simplify to:
det(B) = 1 * ((-21) - 20) - 2 * (0 - (-50)) + 3 * (0 - (30)), which we simplify to:
det(B) = (-41) - 100 - 90 = -231
Method 2
This method it similar to 2 by 2 determinants, and based on opertations with diagonals Again, consider the following matrix, which we will call A:
a | b | c |
d | e | f |
g | h | i |
Then the determinant of this matrix, denoted det(A), is given by:
det(A) = a * e * i + b * f * g + c * d * h - c * e * g - f * h * a - i * b * d
Note how three top-right to bottom-left diagonals are positive
a | ||
e | ||
i |
b | ||
f | ||
g |
c | ||
d | ||
h |
Top-left to bottom-right are negative
c | ||
e | ||
g |
a | ||
f | ||
h |
b | ||
d | ||
i |
Consider the same example as in method 2: matrix, which we will call B:
1 | 2 | 3 |
0 | -3 | 5 |
-10 | 4 | 7 |
det(B) is given by the formula above. We apply the formula below:
det(B) = 1 * (-3) * 7 + 2 * 5 * (-10) + 3 * 0 * 4 - 3 * (-3) * (-10) - 5 * 4 * 1 - 7 * 2 * 0, which we simplify to:
det(B) = -21 - 100 + 0 - 90 - 20 - 0 = -231, same, as in method 1
More information:
- Determinant of a Matrix on MathIsFun
- 3x3 Determinant calculator
- Determinant on Wikipedia