The determinant of a 3×3 matrix is a scalar value that can be computed using a specific formula. Here’s the formula and some solved examples:
Formula
The determinant of a 3×3 matrix A can be computed using the following formula:
|A| = a(ei – fh) – b(di – fg) + c(dh – eg)
where a, b, c, d, e, f, g, h, and i are the elements of matrix A.
Solved Examples
Example 1
Find the determinant of the following matrix:
| 1 2 3 |
| 4 5 6 |
| 7 8 9 |
Using the formula:
|A| = 1(5_9 – 6_8) – 2(4_9 – 6_7) + 3(4_8 – 5_7)
|A| = 1(45 – 48) – 2(36 – 42) + 3(32 – 35)
|A| = 1(-3) – 2(-6) + 3(-3)
|A| = -3 + 12 – 9
|A| = 0
Example 2
Find the determinant of the following matrix:
| 2 0 1 |
| 3 2 1 |
| 1 1 0 |
Using the formula:
|A| = 2(2_0 – 1_1) – 0(3_0 – 1_1) + 1(3_1 – 2_1)
|A| = 2(0 – 1) – 0(0 – 1) + 1(3 – 2)
|A| = 2(-1) – 0(-1) + 1(1)
|A| = -2 + 1
|A| = -1
Example 3
Find the determinant of the following matrix:
| 1 3 2 |
| 4 1 3 |
| 2 5 1 |
Using the formula:
|A| = 1(1_1 – 3_5) – 3(4_1 – 3_2) + 2(4_5 – 1_2)
|A| = 1(1 – 15) – 3(4 – 6) + 2(20 – 2)
|A| = 1(-14) – 3(-2) + 2(18)
|A| = -14 + 6 + 36
|A| = 28