Determinants

Subtopic: Matricies

The determinant is a single number that captures essential information about a square matrix—whether it's invertible, how it scales volumes, and the product of its eigenvalues. This page covers computation methods for 2⨉2 and 3⨉3matrices, key properties, and geometric interpretation.

Introduction

Given a square matrix, can you compress its essential character into a single number? The determinant does exactly that. It tells you whether the matrix is invertible (nonzero determinant) and how the associated transformation scales volumes.

A determinant of 2means the transformation doubles volumes. A determinant of -1 means it preserves volume but flips orientation. A determinant of 0 means it collapses space to a lower dimension.

2×2 Determinant

For a 2⨉2 matrix:

det(a)=a*d−b*c

Geometrically, |a*d-b*c| is the area of the parallelogram spanned by the column vectors. The sign indicates orientation: positive preserves, negative flips.

3×3 Determinant

For a 3⨉3 matrix, expand along the first row:

det(a)=a*(e*i−f*h)−b*(d(i)−f*g)+c*(d(h)−e*g)

Or use the diagonal trick (Sarrus' rule): add products of main diagonals, subtract products of anti-diagonals.

General Definition

For an n⨉n matrix A, the determinant is defined by cofactor expansion:

det(A)=(∑_j=1^n)(−1)*(a_1*j)*det((A_1*j))

where (A_1*j) is the (n-1)⨉(n-1) matrix obtained by deleting row 1 and column j.

This recursive definition reduces an n⨉n determinant to (n-1)⨉(n-1) determinants, eventually reaching 2⨉2.

Worked Example

Compute the determinant of:

A=([1,2,3],[0,4,5],[0,0,6])

This is upper triangular. For triangular matrices, det = product of diagonal entries:

det(A)=1⋅4⋅6=24

Verification: Expanding along row 1 would give the same answer with more work.

Key Properties

Multiplicativity

det(A*B)=det(A)*det(B)

Scaling effects multiply. If A doubles volumes and B triples them, A*B multiplies by 6.

Transpose

det(AT)=det(A)

Rows and columns contribute equally to the determinant.

Inverse

det(A(−1))=1/det(A)

If A scales by k, its inverse scales by 1/k .

Row Operations

• Swapping rows: multiplies determinant by -1

• Multiplying a row by c: multiplies determinant by c

• Adding a multiple of one row to another: determinant unchanged

Triangular Matrices

For triangular matrices (upper or lower), the determinant equals the product of diagonal entries.

When is det(A) = 0?

A matrix has zero determinant exactly when:

• A is singular (not invertible)

• The columns are linearly dependent

• The rows are linearly dependent

• At least one eigenvalue is zero

• The transformation collapses space (maps to lower dimension)

Connection to Eigenvalues

The determinant equals the product of eigenvalues:

det(A)=(λ_1)*(λ_2)⋯(λ_n)

Combined with trace = sum of eigenvalues, these provide quick checks on eigenvalue computations.

Geometric Interpretation

The absolute value |det(A)| is the factor by which A scales n-dimensional volumes.

In 2D: |det(A)| scales areas. In 3D: |det(A)| scales volumes.

The sign indicates orientation: det(A)>0 preserves orientation, det(A)<0 reverses it (like a reflection).

Applications

In solving systems: Cramer's rule uses determinants (though it's computationally inefficient).

In calculus: The Jacobian determinant gives the volume scaling factor for change of variables in multiple integrals.

In eigenvalue problems: det(A-λ*l)=0 defines the characteristic polynomial.

In differential equations: The Wronskian (a determinant) tests linear independence of solutions.

Summary

The determinant is a scalar that encodes whether a matrix is invertible and how it scales volumes. For a 2×2 matrix, det(a)=a*d−b*c. For larger matrices, compute it via cofactor expansion or by using determinant properties. Key facts: det(A*B)=det(A)*det(B), det(AT)=det(A), the determinant equals the product of eigenvalues, and a zero determinant means the matrix is singular. The sign of the determinant indicates whether orientation is preserved or reversed.