1. Why Determinants Are Useful
Determinants are more than just a number you calculate. They help you understand whether a matrix is invertible, how a system of equations behaves, how shapes transform, and whether vectors are linearly independent. These applications appear throughout algebra, geometry, and even physics.
2. Application 1: Checking If a Matrix Is Invertible
A matrix is invertible if and only if its determinant is non-zero.
|A| \neq 0 \Rightarrow A^{-1} \text{ exists}
|A| = 0 \Rightarrow A \text{ is singular (no inverse)}
This is one of the most common uses of determinants in linear algebra.
2.1. Example
A = \begin{bmatrix} 2 & 3 \\ 4 & 6 \end{bmatrix}, \quad |A| = (2)(6) - (3)(4) = 0
Determinant is zero → not invertible.
3. Application 2: Solving Linear Equations Using Cramer's Rule
Cramer's Rule provides a way to solve a system of linear equations using determinants.
For a system:
AX = B
If \( |A| \neq 0 \), each variable is given by:
x_i = \frac{|A_i|}{|A|}
where \( A_i \) is the matrix formed by replacing the \( i \)-th column of \( A \) with the constants column.
3.1. Example (2×2 System)
System:
2x + 3y = 5 \\ 4x + y = 6
Coefficient matrix:
A = \begin{bmatrix} 2 & 3 \\ 4 & 1 \end{bmatrix}
Compute \( |A| \):
|A| = (2)(1) - (3)(4) = -10
Now replace columns to find determinants for x and y.
4. Application 3: Area of a Triangle Using Coordinates
In coordinate geometry, the area of a triangle with vertices \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) is given by:
\text{Area} = \frac{1}{2} \left| \begin{matrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{matrix} \right|
This formula comes directly from a determinant.
4.1. Example
Points: (1, 2), (3, 4), (5, 1)
Area =
\frac{1}{2} \left| \begin{matrix} 1 & 2 & 1 \\ 3 & 4 & 1 \\ 5 & 1 & 1 \end{matrix} \right|
5. Application 4: Testing Linear Dependence of Vectors
The determinant helps identify if vectors are linearly dependent.
For 3 vectors in \( \mathbb{R}^3 \), place them as rows (or columns) of a 3×3 matrix:
|A| = 0 \Rightarrow \text{vectors are linearly dependent}
|A| \neq 0 \Rightarrow \text{vectors are linearly independent}
5.1. Example
Vectors:
v_1 = (1, 2, 3), \; v_2 = (2, 4, 6), \; v_3 = (3, 6, 9)
Matrix formed from these rows has determinant 0 → dependent.
6. Application 5: Volume of a Parallelepiped (Using 3×3 Determinant)
The volume formed by three vectors in space is the absolute value of the determinant of the matrix formed by those vectors.
V = | \det(A) |
This is used in physics and 3D geometry.
7. Application 6: Behavior of Transformations
In transformations (like scaling, rotation, shearing), the determinant of the transformation matrix tells us two things:
- Area/Volume scaling factor
- Orientation change (positive = preserved, negative = flipped)
This idea is heavily used in computer graphics, robotics, and geometry.
8. Application 7: Checking Consistency of a System
In linear algebra, determinants help distinguish whether a system of equations has:
- a unique solution
- no solution
- infinitely many solutions
This is linked to the rank of a matrix and the determinant of the coefficient matrix.