Singular Matrix

Singular Matrix
Noninvertible Matrix

A square matrix which does not have an inverse. A matrix is singular if and only if its determinant is zero.

Example: Matrix [[2,6],[1,3]] is singular since its determinant |2 6 / 1 3| = 2·3 − 6·1 = 0

See also

Key Formula

\det(A) = 0 \iff A \text{ is singular}

Where:

$A$ = A square matrix (same number of rows and columns)
$\det(A)$ = The determinant of matrix A
$0$ = A determinant of zero means the matrix has no inverse

Worked Example

Problem: Determine whether the matrix A is singular: A = [[2, 6], [1, 3]].

Step 1: Write down the formula for the determinant of a 2×2 matrix.

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

Step 2: Identify the entries: a = 2, b = 6, c = 1, d = 3.

A = \begin{pmatrix} 2 & 6 \\ 1 & 3 \end{pmatrix}

Step 3: Compute the determinant by substituting into the formula.

\det(A) = (2)(3) - (6)(1) = 6 - 6 = 0

Step 4: Since the determinant equals zero, the matrix is singular and has no inverse.

\det(A) = 0 \implies A \text{ is singular}

Answer: Matrix A is singular because its determinant is 0.

Another Example

This example extends to a 3×3 matrix and shows that linear dependence among rows is the geometric reason a matrix is singular.

Problem: Determine whether the 3×3 matrix B is singular: B = [[1, 2, 3], [4, 5, 6], [7, 8, 9]].

Step 1: Use cofactor expansion along the first row to compute the determinant of a 3×3 matrix.

\det(B) = 1\det\begin{pmatrix}5 & 6\\8 & 9\end{pmatrix} - 2\det\begin{pmatrix}4 & 6\\7 & 9\end{pmatrix} + 3\det\begin{pmatrix}4 & 5\\7 & 8\end{pmatrix}

Step 2: Evaluate each 2×2 determinant.

\det\begin{pmatrix}5&6\\8&9\end{pmatrix} = 45 - 48 = -3, \quad \det\begin{pmatrix}4&6\\7&9\end{pmatrix} = 36 - 42 = -6, \quad \det\begin{pmatrix}4&5\\7&8\end{pmatrix} = 32 - 35 = -3

Step 3: Substitute back and simplify.

\det(B) = 1(-3) - 2(-6) + 3(-3) = -3 + 12 - 9 = 0

Step 4: The determinant is zero, so B is singular. Notice that the third row is the sum of the first two rows (row 1 + row 2 = [1+4, 2+5, 3+6] = [5, 7, 9]... actually row 3 = 2·row2 − row1). This linear dependence among rows always produces a singular matrix.

\text{Row 3} = 2 \cdot \text{Row 2} - \text{Row 1}: \quad 2(4,5,6) - (1,2,3) = (7,8,9) \checkmark

Answer: Matrix B is singular (det(B) = 0) because its rows are linearly dependent.

Frequently Asked Questions

What is the difference between a singular matrix and a non-singular matrix?

A singular matrix has a determinant of zero and no inverse, while a non-singular (invertible) matrix has a nonzero determinant and does have an inverse. If you try to solve a system of equations using a singular coefficient matrix, you will either get no solution or infinitely many solutions — never a unique one.

Can a non-square matrix be singular?

The term 'singular' is defined only for square matrices, because determinants and matrix inverses are defined only for square matrices. Non-square matrices cannot be inverted in the usual sense, so the concept does not apply to them directly.

How do you know if a matrix is singular without computing the determinant?

You can row-reduce the matrix to echelon form. If any row becomes all zeros, the matrix is singular. Equivalently, if the columns (or rows) are linearly dependent — meaning one can be written as a combination of the others — the matrix is singular. A zero row or column also guarantees singularity.

Singular Matrix vs. Invertible (Non-Singular) Matrix

	Singular Matrix	Invertible (Non-Singular) Matrix
Determinant	det(A) = 0	det(A) ≠ 0
Inverse exists?	No	Yes: A⁻¹ exists
System Ax = b	No unique solution (0 or ∞ solutions)	Exactly one unique solution: x = A⁻¹b
Row reduction	At least one row of zeros in echelon form	All pivots present; reduces to identity
Rows/columns	Linearly dependent	Linearly independent
Rank	Rank < n (not full rank)	Rank = n (full rank)

Why It Matters

Singular matrices appear whenever you solve systems of linear equations — if the coefficient matrix is singular, the system has no unique solution, which signals either no solution or infinitely many. In applications such as computer graphics, engineering, and data science, checking whether a matrix is singular is a crucial first step before attempting to invert it. Understanding singularity also connects to eigenvalues: a matrix is singular if and only if zero is one of its eigenvalues.

Common Mistakes

Mistake: Assuming a matrix with a row or column of zeros is the only way to be singular.

Correction: A matrix can be singular without any zero entries. For example, [[2, 6], [1, 3]] has all nonzero entries but is singular because its rows are proportional (row 1 = 2 × row 2), giving det = 0.

Mistake: Trying to compute the inverse of a singular matrix by using the adjugate formula and dividing by the determinant.

Correction: Since the determinant is zero, dividing by it is undefined. Always check that det(A) ≠ 0 before applying the inverse formula A⁻¹ = (1/det(A)) · adj(A).

Related Terms

Determinant — Zero determinant defines a singular matrix
Inverse of a Matrix — Singular matrices lack an inverse
Invertible Matrix — The opposite: a non-singular matrix
Square Matrix — Singularity applies only to square matrices
If and Only If — Logical equivalence used in the definition
Zero — The determinant value that indicates singularity
Linear Dependence — Dependent rows/columns cause singularity
Eigenvalue — Singular iff zero is an eigenvalue