The determinant is a scalar value associated with every square matrix. It provides crucial information about the matrix — whether it has an inverse, the area/volume of geometric figures, and the solvability of linear systems.
---
Key Concepts
Determinant of a 2 x 2 Matrix:
If A = [[a, b], [c, d]], then det(A) = |A| = ad - bc.
Determinant of a 3 x 3 Matrix (Expansion along first row):
|A| = a11 · C11 + a12 · C12 + a13 · C13
where Cij = (-1)i+j · Mij (cofactor), and Mij is the minor (determinant of the 2x2 matrix obtained by deleting row i and column j).
- 1.Properties of Determinants:
- 2.|AT| = |A|
- 3.If two rows (or columns) are identical, |A| = 0.
- 4.If a row/column is multiplied by k, |A| is multiplied by k.
- 5.|kA| = kn · |A| for an n x n matrix.
- 6.Swapping two rows changes the sign of the determinant.
- 7.Adding a scalar multiple of one row to another does not change the determinant.
- 8.|AB| = |A| · |B|.
Singular Matrix: A matrix is singular if |A| = 0. A singular matrix has no inverse.
- Cofactor and Adjoint:
- Cofactor Cij = (-1)i+j · Mij.
- Adjoint (adj A): Transpose of the cofactor matrix.
- Inverse: A-1 = adj(A) / |A|, provided |A| is not 0.
Consistency of Linear Systems (Cramer's Rule):
For AX = B: x = |A1|/|A|, y = |A2|/|A|, z = |A3|/|A|, where |Ai| is obtained by replacing the i-th column with the B column.
Area of Triangle: Area = (1/2) |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)| using the determinant formula.
---
Worked Examples
Evaluate the determinant [[3, 2], [1, 4]].
|A| = 3 · 4 - 2 · 1 = 12 - 2 = 10.
Evaluate [[1, 2, 3], [0, 1, 4], [5, 6, 0]].
Expanding along row 1:
= 1 · (1 · 0 - 4 · 6) - 2 · (0 · 0 - 4 · 5) + 3 · (0 · 6 - 1 · 5)
= 1 · (0 - 24) - 2 · (0 - 20) + 3 · (0 - 5)
= -24 + 40 - 15 = 1.
Find the minor and cofactor of element 3 in the matrix [[1, 2, 3], [0, -1, 4], [2, 5, 0]].
Element 3 is at position (1, 3). Minor M13 = |[[0,-1],[2,5]]| = 0 · 5 - (-1) · 2 = 2.
Cofactor C13 = (-1)1+3 · 2 = (+1) · 2 = 2.
If A = [[2, 3], [1, 4]], find A-1.
|A| = 2 · 4 - 3 · 1 = 5. Cofactor matrix: C11 = 4, C12 = -1, C21 = -3, C22 = 2.
adj(A) = [[4, -3], [-1, 2]].
A-1 = (1/5) · [[4, -3], [-1, 2]].
Find the area of a triangle with vertices A(1, 0), B(6, 0), C(4, 3).
Area = (1/2)|1(0 - 3) + 6(3 - 0) + 4(0 - 0)| = (1/2)|(-3 + 18 + 0)| = (1/2) · 15 = 7.5 sq units.
Solve using Cramer's rule: 2x + y = 5, x - y = 1.
|A| = |[[2,1],[1,-1]]| = -2 - 1 = -3.
|A1| = |[[5,1],[1,-1]]| = -5 - 1 = -6. x = -6/-3 = 2.
|A2| = |[[2,5],[1,1]]| = 2 - 5 = -3. y = -3/-3 = 1.
Show that [[a, b, c], [a+2x, b+2y, c+2z], [x, y, z]] = 0.
Row 2 = Row 1 + 2 · (Row 3). Since one row is a linear combination of others, the determinant is 0.
---
Common mistakes
- Forgetting the alternating sign pattern (-1)i+j when computing cofactors.
- Expanding the determinant along the wrong row/column without applying the sign pattern correctly.
- Not dividing each element by |A| when computing the inverse from the adjoint.
---
Summary
Determinants are powerful tools for computing inverses, areas, and solving linear equations. Their properties allow simplification before evaluation. Cramer's rule provides an elegant formula for solutions of linear systems.