Concept

Singular Matrix

A singular matrix is a matrix whose determinant is zero.

A is a singular matrix ⇕ det (A) = 0

Since its determinant is zero, a singular matrix has no inverse.

Example

Consider a

2 \times 2

matrix

A .

A = [2163]

Whether a matrix is singular depends on its determinant.

∣ A ∣ = det [2163]

▼

Simplify right-hand side

Calculate the determinant of a $2 \times 2$ matrix

∣ A ∣ = 2 (3) - 6 (1)

∣ A ∣ = 6 - 6

Subtract term

∣ A ∣ = 0

Since its determinant is zero,

A

is a singular matrix.

Why Do We Calculate the Determinant

A link between the determinant and the the inverse of a non-singular matrix can be illustrated in the formula for the inverse of a

2 \times 2

A = [a c b d] ⇓ A^{- 1} = \frac{1}{det ( A )} [d - c - b a]

The determinant is in a denominator. Therefore, the formula is well defined if and only if the determinant is not zero.