IsSingular

Tests if a square matrix is singular.

Syntax

• IsSingular(A)

  • A is a square matrix

Description

If A is a square matrix of size n, then IsSingular(A) returns true iff A is singular.

The following are equivalent:

• A is NOT singular.

• det(A) ≠ 0.

• rank(A) = n.

• The columns of A are linearly independent.

• The rows of A are linearly independent.

• A is invertible.

• The matrix equation AX = Y has a unique solution X for every Y.

• 0 ∉ spectrum(A).

Examples

A ≔ ❨❨5, 3, 2, 1❩, ❨−2, 5, 1, −2❩, ❨1, −2, 3, 1❩, ❨4, 6, 6, 0❩❩

⎛ 5   3   2   1⎞
⎜−2   5   1  −2⎟
⎜ 1  −2   3   1⎟
⎝ 4   6   6   0⎠

IsSingular(A)

true

'(det(A), rank(A), #cols(ColumnSpaceBasis(A)), IsZero(min(spectrum(A) @ abs)))

0
3
3
true

See also

• det

• rank

• inv

• spectrum (eigenvalues)