Determinants

Summary

The determinant is a scalar associated to a square matrix. It measures signed volume scaling of the linear map, vanishes exactly when the matrix is singular, and appears in the eigenvalue equation $det (A - λ I) = 0$ .

Prerequisites

Matrices and Row Reduction. Hub: Linear Algebra.

Definition / Statement

For a square matrix $A \in R^{n \times n}$ , the determinant $det (A)$ (also written $| A |$ ) is the unique alternating multilinear function of the columns with $det (I) = 1$ . Computationally it is defined recursively by cofactor expansion, or via row reduction.

Objects and Dimensions

Object	Meaning	Dimensions
$A$	square matrix	$n \times n$
$det (A)$	determinant	scalar
$adj (A)$	adjugate (transpose of cofactor matrix)	$n \times n$

Notation

Symbol	Meaning
$det (A)$ , $	A
$C_{i j}$	$(i, j)$ -cofactor
$A_{i j}$	minor deleting row $i$ , column $j$

Conditions / Assumptions

$A$ must be square for $det (A)$ to be defined in this sense.
$det (A) \neq 0$ if and only if $A$ is invertible.
Over $R$ , $| det (A) |$ is the volume scaling factor of the unit cube under $A$ .

Matrix / Vector Form

$2 \times 2$

det (\begin{matrix} a & b \\ c & d \end{matrix}) = a d - b c .

$3 \times 3$ (cofactor expansion along the first row)

det (\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}) = a_{11} (a_{22} a_{33} - a_{23} a_{32}) - a_{12} (a_{21} a_{33} - a_{23} a_{31}) + a_{13} (a_{21} a_{32} - a_{22} a_{31}) .

Product and transpose

det (A B) = det (A) det (B), det (A^{T}) = det (A) .

Inverse (when $det (A) \neq 0$ )

A^{- 1} = \frac{1}{det (A)} adj (A) .

Row-reduction rules (sign and scale carefully):

Swapping two rows multiplies $det$ by $- 1$ .
Multiplying a row by $c$ multiplies $det$ by $c$ .
Adding a multiple of one row to another leaves $det$ unchanged.
Determinant of a triangular matrix is the product of diagonal entries.

Geometric Interpretation

$det (A) = 0$ : columns lie in a lower-dimensional flat; volume collapses.
$det (A) > 0$ : orientation-preserving map; $det (A) < 0$ : orientation-reversing.

Worked Example

det (\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}) = 1 \cdot 4 - 2 \cdot 3 = - 2 \neq 0,

so the matrix is invertible. For

B = (\begin{matrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{matrix}), det (B) = 1 \cdot 4 \cdot 6 = 24.

Common Mistakes

Using $a d + b c$ instead of $a d - b c$ for $2 \times 2$ .
Applying the $2 \times 2$ pattern naively to $3 \times 3$ without cofactors (the “diagonal trick” needs care with signs).
Concluding $det (A + B) = det (A) + det (B)$ (false in general).
Computing determinants of non-square matrices.

Connections

Related: Matrices and Row Reduction, Eigenvalues and Eigenvectors, Systems of Linear Equations
Next: Eigenvalues and Eigenvectors
Volume links: bases and parallelepipeds in Vector Spaces and Bases

References

Determinant properties and computations follow MIT 18.06.^[1]

MIT OpenCourseWare, 18.06 Linear Algebra, https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/ ↩︎