Determinants of Matrices

Master the calculation and properties of matrix determinants

What is a Determinant?

A determinant is a special number (scalar) that can be calculated from the elements of a square matrix. It provides important information about the matrix and is essential for solving systems of linear equations, finding matrix inverses, and understanding linear transformations.

Determinant Definition:

The determinant is a number uniquely associated with a square matrix that characterizes certain properties of the matrix and the system it represents.

Key Applications:
  • Determining if a matrix is invertible
  • Solving systems of linear equations (Cramer's rule)
  • Computing matrix inverses
  • Understanding volume scaling in linear transformations

Determinant Notation

Common Notations

For a matrix \(A\), the determinant can be denoted in several ways:

  • \(\det(A)\) - functional notation
  • \(|A|\) - vertical bar notation
  • \(\begin{vmatrix}a & b \\ c & d\end{vmatrix}\) - vertical bars around matrix elements

2×2 Determinants

Formula for 2×2 Matrices

2×2 Determinant Formula:

For a 2×2 matrix, the determinant is calculated as the product of the diagonal elements minus the product of the off-diagonal elements:

2×2 Determinant:
\(\displaystyle \det\begin{bmatrix}a & b \\ c & d\end{bmatrix} = \begin{vmatrix}a & b \\ c & d\end{vmatrix} = ad - bc\)

Example: 2×2 Determinant

Calculate the Determinant

Given matrix:

\(\displaystyle A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}\)

Calculation:

\(\displaystyle \det(A) = (1)(4) - (2)(3) = 4 - 6 = -2\)

Result: The determinant is \(-2\)

3×3 Determinants

Cofactor Expansion Method

3×3 Determinant Calculation:

For a 3×3 matrix, the determinant is found by expanding along any row or column. Each element is multiplied by its minor (the determinant of the 2×2 submatrix remaining after removing that element's row and column), with alternating signs.

Expanding Along First Row
\(\displaystyle \det\begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix} = a\begin{vmatrix}e & f \\ h & i\end{vmatrix} - b\begin{vmatrix}d & f \\ g & i\end{vmatrix} + c\begin{vmatrix}d & e \\ g & h\end{vmatrix}\)

Simplified form:

\(\displaystyle = a(ei - fh) - b(di - fg) + c(dh - eg)\)
\(\displaystyle = aei - afh - bdi + bfg + cdh - ceg\)

Key Terms

Minor of Element
The determinant of the 2×2 submatrix obtained by removing the element's row and column
Cofactor
The minor multiplied by \((-1)^{i+j}\) where \(i\) and \(j\) are the row and column indices
Sign Pattern
Alternates: +, −, +, −, ... starting with + for position (1,1)

Example: 3×3 Determinant

Calculate a 3×3 Determinant

Given matrix:

\(\displaystyle A = \begin{bmatrix}1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0\end{bmatrix}\)

Expanding along the first row:

\(\displaystyle \det(A) = 1\begin{vmatrix}1 & 4 \\ 6 & 0\end{vmatrix} - 2\begin{vmatrix}0 & 4 \\ 5 & 0\end{vmatrix} + 3\begin{vmatrix}0 & 1 \\ 5 & 6\end{vmatrix}\)

Calculate 2×2 minors:

\(\displaystyle = 1(0 - 24) - 2(0 - 20) + 3(0 - 5)\)
\(\displaystyle = 1(-24) - 2(-20) + 3(-5)\)
\(\displaystyle = -24 + 40 - 15 = 1\)

Result: The determinant is \(1\)

4×4 and Larger Determinants

The same cofactor expansion method applies to larger matrices. For a 4×4 matrix, expand along any row or column to get four 3×3 determinants, which are then computed using the method for 3×3 determinants.

Recursive Approach

For an \(n \times n\) matrix:

  1. Choose any row or column to expand along
  2. For each element, calculate its minor (determinant of the \((n-1) \times (n-1)\) submatrix)
  3. Apply the correct sign (alternating pattern)
  4. Sum all terms
Computational Complexity:

This method becomes computationally expensive for large matrices. For 4×4 or larger matrices, it's more efficient to use row reduction or specialized algorithms.

Properties of Determinants

Key Properties

Property Description Example Impact
Row/Column Swap Swapping two rows or columns changes the sign of the determinant \(\det(A') = -\det(A)\)
Scalar Multiplication Multiplying a row/column by scalar \(k\) multiplies the determinant by \(k\) \(\det(kA) = k^n \det(A)\) for \(n \times n\) matrix
Equal Rows/Columns If two rows or columns are identical, the determinant is zero \(\det(A) = 0\)
Row/Column Operations Adding a multiple of one row/column to another doesn't change the determinant \(\det(A) = \det(A')\)
Proportional Rows/Columns If two rows/columns are proportional, the determinant is zero \(\det(A) = 0\)
Triangular Matrix Determinant equals the product of diagonal elements \(\det(A) = a_{11} \cdot a_{22} \cdot ... \cdot a_{nn}\)

Singular Matrices

Definition

Singular Matrix:

A matrix is singular if its determinant equals zero. A singular matrix has no inverse and represents a degenerate transformation.

Conditions for a Singular Matrix

A matrix is singular (determinant = 0) if any of the following is true:

  • All elements of one row or column are zero
  • Two rows or columns are identical
  • Two rows or columns are proportional (one is a scalar multiple of another)
  • One row or column is a linear combination of other rows or columns
Example of Singular Matrix

Matrix with proportional rows:

\(\displaystyle A = \begin{bmatrix}1 & 2 \\ 2 & 4\end{bmatrix}\)

Calculation:

\(\displaystyle \det(A) = (1)(4) - (2)(2) = 4 - 4 = 0\)

This matrix is singular because Row 2 = 2 × Row 1. The matrix has no inverse.

Determinants of Triangular Matrices

Special Property

Triangular Matrix Determinant:

For a triangular matrix (upper or lower), the determinant equals the product of the diagonal elements.

Why This Works

In an upper triangular matrix, all elements below the diagonal are zero. When expanding the determinant, most terms become zero because they would require multiplying a zero element. Only the product of diagonal elements remains.

Upper Triangular Matrix

Matrix:

\(\displaystyle A = \begin{bmatrix}2 & 3 & 1 \\ 0 & 5 & 4 \\ 0 & 0 & -2\end{bmatrix}\)

Determinant:

\(\displaystyle \det(A) = 2 \cdot 5 \cdot (-2) = -20\)

Key Points to Remember

  • Determinants are only defined for square matrices
  • For 2×2 matrices: \(\det(A) = ad - bc\)
  • For larger matrices: use cofactor expansion method
  • If \(\det(A) = 0\), the matrix is singular and non-invertible
  • Swapping rows/columns changes the sign of the determinant
  • Proportional rows/columns result in \(\det(A) = 0\)
  • For triangular matrices: \(\det(A) = \) product of diagonal elements
  • Determinants help determine if systems of equations have unique solutions
  • Essential for computing matrix inverses using Cramer's rule

Practice Your Skills

Test your understanding of determinants with our interactive calculators:

3×3 Determinant Calculator →

4×4 Determinant Calculator →
































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