Determinants

Definition and Basic Properties

The determinant is a scalar value that can be calculated from the elements of a square matrix. It provides important information about the matrix, such as whether it is invertible.

Notation

For a matrix A, the determinant is denoted det(A) or |A|.

2×2 Determinant

For a 2×2 matrix $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$:

• det(A) = ad - bc

3×3 Determinant

For a 3×3 matrix $A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix}$:

det(A) = a₁₁(a₂₂a₃₃ - a₂₃a₃₂) - a₁₂(a₂₁a₃₃ - a₂₃a₃₁) + a₁₃(a₂₁a₃₂ - a₂₂a₃₁)

This can also be computed using the Sarrus rule (diagonals method):

1. Extend the matrix by copying the first two columns to the right
2. Multiply along the diagonals going down-right and add these products
3. Multiply along the diagonals going down-left and subtract these products

n×n Determinant

For larger matrices, determinants can be calculated using:

1. Cofactor expansion: Select a row or column and expand along it
2. Row/column operations: Transform the matrix to upper triangular form

Properties of Determinants

1. Transpose: det(A) = det(A^T)

2. Scalar multiplication: det(kA) = k^n · det(A) where n is the size of the matrix

3. Product: det(AB) = det(A) · det(B)

4. Inverse: If A is invertible, det(A^(-1)) = 1/det(A)

5. Row/Column operations:
   • Swapping two rows/columns multiplies the determinant by -1
   • Multiplying a row/column by a scalar k multiplies the determinant by k
   • Adding a multiple of one row/column to another doesn’t change the determinant

6. Triangular matrices: For triangular matrices (upper or lower), the determinant is the product of the diagonal elements

7. Singular matrices: A matrix is singular (non-invertible) if and only if its determinant is zero

Calculation Methods

Cofactor Expansion (Laplace Expansion)

The determinant can be calculated by expanding along any row or column:

det(A) = Σⱼ (-1)^(i+j) · aᵢⱼ · Mᵢⱼ

Where:

• aᵢⱼ is the element in row i, column j
• Mᵢⱼ is the minor (determinant of the submatrix formed by removing row i and column j)
• (-1)^(i+j) · Mᵢⱼ is the cofactor of aᵢⱼ

Row Reduction (Gaussian Elimination)

1. Convert the matrix to upper triangular form using row operations
2. The determinant is the product of the diagonal elements, adjusted by the factor (-1)^s where s is the number of row swaps

Special Cases

1. Diagonal matrix: det(diag(d₁, d₂, …, dₙ)) = d₁ · d₂ · … · dₙ

2. Identity matrix: det(I) = 1

3. Zero matrix: det(0) = 0

4. Matrix with a zero row/column: det(A) = 0

5. Matrix with two identical rows/columns: det(A) = 0

6. Matrix with a row/column that’s a linear combination of others: det(A) = 0

Example Calculations

2×2 Determinant

For $A = \begin{bmatrix} 3 & 2 \ 4 & 5 \end{bmatrix}$:

• det(A) = 3×5 - 2×4 = 15 - 8 = 7

3×3 Determinant

For $A = \begin{bmatrix} 3 & 1 & 2 \ -3 & -4 & 6 \ 0 & -3 & 8 \end{bmatrix}$:

Using cofactor expansion along the first row:

• det(A) = 3·det($\begin{bmatrix} -4 & 6 \ -3 & 8 \end{bmatrix}$) - 1·det($\begin{bmatrix} -3 & 6 \ 0 & 8 \end{bmatrix}$) + 2·det($\begin{bmatrix} -3 & -4 \ 0 & -3 \end{bmatrix}$)
• det(A) = 3·(-4·8 - 6·(-3)) - 1·(-3·8 - 6·0) + 2·(-3·(-3) - (-4)·0)
• det(A) = 3·(-32 + 18) - 1·(-24) + 2·(9)
• det(A) = 3·(-14) + 24 + 18
• det(A) = -42 + 24 + 18 = 0

4×4 Determinant

For a 4×4 matrix, it’s often easiest to use row reduction:

1. Convert to upper triangular form using Gaussian elimination
2. Multiply the diagonal elements

Strategies for Determinant Exercises

1. Choose the easiest expansion path: Look for rows or columns with many zeros

2. Use properties to simplify:
   • Factor out common terms from rows/columns
   • Add/subtract rows to create zeros
3. Recognize patterns:
   • Triangular matrices: product of diagonal elements
   • Matrices with repeated rows/columns: determinant is zero
4. For parametric problems:
   • Calculate the determinant keeping the parameter(s)
   • Analyze when the resulting expression equals zero

Practice Examples

Example 1: Parameter Determinant

For $A = \begin{bmatrix} a & 3a & 2a \ -a & -4a & 6 \ a-1 & 2a & a+1 \end{bmatrix}$:

Factor out a from the first row and -a from the second row: $\begin{bmatrix} a & 3a & 2a \ -a & -4a & 6 \ a-1 & 2a & a+1 \end{bmatrix} = a \cdot \begin{bmatrix} 1 & 3 & 2 \ -1 & -4 & \frac{6}{-a} \ \frac{a-1}{a} & 2 & \frac{a+1}{a} \end{bmatrix}$

The determinant is a³ times the determinant of the new matrix. Continue using cofactor expansion.

Example 2: Vandermonde Determinant

For $A = \begin{bmatrix} 1 & 1 & 1 \ x & y & z \ x^2 & y^2 & z^2 \end{bmatrix}$:

This is a Vandermonde matrix with determinant (z-y)(z-x)(y-x)

Example 3: Using Row Operations

For $\begin{bmatrix} 1 & 3 & -2 & -1 \ 0 & 4 & 2 & 5 \ 3 & 1 & 0 & 2 \ -4 & 6 & -5 & 2 \end{bmatrix}$:

Use row operations to create more zeros and transform the matrix to upper triangular form.

Exercises

Calculate the following determinants:

1. $\begin{vmatrix} 3 & 2 \ 4 & 5 \end{vmatrix}$

2. $\begin{vmatrix} -4 & 6 \ 5 & -1 \end{vmatrix}$

3. $\begin{vmatrix} 0 & -5 \ 4 & 8 \end{vmatrix}$

4. $\begin{vmatrix} 3 & 6 \ 9 & 18 \end{vmatrix}$

5. $\begin{vmatrix} -4 & 1 \ 8 & -2 \end{vmatrix}$

6. $\begin{vmatrix} 3 & 1 & 2 \ -3 & -4 & 6 \ 0 & -3 & 8 \end{vmatrix}$

7. $\begin{vmatrix} 3 & -1 & 2 \ 71 & -47 & 61 \ 6 & -2 & 4 \end{vmatrix}$

8. $\begin{vmatrix} a & 3a & 2a \ -a & -4a & 6 \ a-1 & 2a & a+1 \end{vmatrix}$

9. $\begin{vmatrix} 1 & 1 & 1 \ x & y & z \ x^2 & y^2 & z^2 \end{vmatrix}$

10. $\begin{vmatrix} 1 & 3 & -2 & -1 \ 0 & 4 & 2 & 5 \ 3 & 1 & 0 & 2 \ -4 & 6 & -5 & 2 \end{vmatrix}$

11. $\begin{vmatrix} 1 & 4 & 3 & \frac{1}{2} \ 2 & 1 & 0 & 6 \ -2 & 5 & 0 & -3 \ 4 & \frac{1}{5} & 5 & \frac{2}{3} \end{vmatrix}$

12. $\begin{vmatrix} 1 & 3 & 2 & 5 \ 1 & 4 & -2 & 6 \ 2 & 7 & 0 & 11 \ -41 & 6 & 15 & 0 \end{vmatrix}$

13. $\begin{vmatrix} 1 & 4 & -3 & 2 \ 2 & 1 & 1 & 3 \ -2 & 1 & 0 & -2 \ 1 & 6 & -2 & 3 \end{vmatrix}$

14. $\begin{vmatrix} 1 & 0 & 0 & 0 \ 10 & 1 & 100 & 1000 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{vmatrix}$