Determinant

Q: What is the difference between a determinant and absolute value?

Although the vertical-bar notation $|A|$ looks identical to absolute value notation, a determinant and an absolute value are different operations. Absolute value always gives a non-negative result, while a determinant can be positive, negative, or zero. The determinant is computed from a matrix, whereas absolute value is applied to a single real number.

Determinant

A single number obtained from a matrix that reveals a variety of the matrix's properties. Determinants of small matrices are written and evaluated as shown below. Determinants may also be found using expansion by cofactors.

Note: Although a determinant looks like an absolute value it is not. The determinant of a matrix may be negative or positive.

Key Formula

\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc

Where:

$a, b, c, d$ = The four entries of a 2×2 matrix
$ad$ = Product of the main diagonal entries (top-left × bottom-right)
$bc$ = Product of the off-diagonal entries (top-right × bottom-left)

Worked Example

Problem: Find the determinant of the 2×2 matrix A = [[5, 3], [2, 4]].

Step 1: Write the determinant using the vertical-bar notation.

\det(A) = \begin{vmatrix} 5 & 3 \\ 2 & 4 \end{vmatrix}

Step 2: Apply the 2×2 determinant formula: multiply the main diagonal entries.

ad = 5 \times 4 = 20

Step 3: Multiply the off-diagonal entries.

bc = 3 \times 2 = 6

Step 4: Subtract the off-diagonal product from the main diagonal product.

\det(A) = 20 - 6 = 14

Answer: The determinant of A is 14.

Another Example

This example extends to a 3×3 matrix and demonstrates expansion by cofactors, which is the standard method for matrices larger than 2×2.

Problem: Find the determinant of the 3×3 matrix B = [[2, 1, 3], [0, 4, 1], [1, 0, 2]] using expansion along the first row.

Step 1: Write the expansion by cofactors along the first row. Each entry in the first row is multiplied by its cofactor (the signed minor).

\det(B) = 2\begin{vmatrix}4 & 1\\0 & 2\end{vmatrix} - 1\begin{vmatrix}0 & 1\\1 & 2\end{vmatrix} + 3\begin{vmatrix}0 & 4\\1 & 0\end{vmatrix}

Step 2: Evaluate the first 2×2 determinant.

\begin{vmatrix}4 & 1\\0 & 2\end{vmatrix} = (4)(2) - (1)(0) = 8

Step 3: Evaluate the second 2×2 determinant.

\begin{vmatrix}0 & 1\\1 & 2\end{vmatrix} = (0)(2) - (1)(1) = -1

Step 4: Evaluate the third 2×2 determinant.

\begin{vmatrix}0 & 4\\1 & 0\end{vmatrix} = (0)(0) - (4)(1) = -4

Step 5: Combine all three terms with their signs.

\det(B) = 2(8) - 1(-1) + 3(-4) = 16 + 1 - 12 = 5

Answer: The determinant of B is 5.

Frequently Asked Questions

What does a determinant of zero mean?

A determinant of zero means the matrix is singular — it has no inverse. Geometrically, it means the transformation defined by the matrix collapses space into a lower dimension (for example, squashing a plane down to a line). In a system of linear equations, a zero determinant indicates the system either has no solution or infinitely many solutions.

Can you take the determinant of a non-square matrix?

No. The determinant is defined only for square matrices (same number of rows and columns). A 2×3 or 4×2 matrix does not have a determinant.

What is the difference between a determinant and absolute value?

Although the vertical-bar notation

|A|

looks identical to absolute value notation, a determinant and an absolute value are different operations. Absolute value always gives a non-negative result, while a determinant can be positive, negative, or zero. The determinant is computed from a matrix, whereas absolute value is applied to a single real number.

Determinant vs. Matrix

	Determinant	Matrix
What it is	A single number (scalar) derived from a square matrix	A rectangular array of numbers arranged in rows and columns
Notation	det(A) or \|A\| using vertical bars	Square brackets [ ] or parentheses ( )
Size requirement	Defined only for square matrices (n × n)	Can be any size (m × n)
Result	A single number that can be positive, negative, or zero	A structured collection of numbers, not a single value
Purpose	Tests invertibility, measures area/volume scaling, solves systems via Cramer's rule	Represents data, linear transformations, or systems of equations

Why It Matters

The determinant appears throughout algebra, calculus, and physics. You use it to determine whether a system of linear equations has a unique solution, to find the inverse of a matrix, and to compute cross products in vector calculus. In geometry, the absolute value of a determinant gives the scale factor by which a linear transformation stretches or compresses areas (in 2D) or volumes (in 3D).

Common Mistakes

Mistake: Subtracting in the wrong order for a 2×2 determinant — computing bc − ad instead of ad − bc.

Correction: Always start with the main diagonal (top-left to bottom-right) product and subtract the off-diagonal product: det = ad − bc.

Mistake: Forgetting the alternating sign pattern (+, −, +, −, …) when expanding a 3×3 or larger determinant by cofactors.

Correction: The sign of each cofactor follows a checkerboard pattern starting with + in the top-left corner. For entry in row i, column j, the sign is (−1)^(i+j). Missing a sign will give the wrong answer.

Related Terms

Matrix — The structure a determinant is computed from
Expansion by Cofactors — Method for evaluating larger determinants
Absolute Value — Shares notation but is a different operation
Inverse of a Matrix — Exists only when the determinant is nonzero
Cramer's Rule — Solves linear systems using determinants
Square Matrix — Determinants are defined only for square matrices
Negative Number — Determinants can be negative
Positive Number — Determinants can be positive