Determinant of a Matrix

Every square matrix is associated with a single real (or complex) number called its determinant.

• If A is a square matrix, its determinant is denoted by |A| or det(A).

• Only square matrices (order \[n \times n\]) have determinants; rectangular matrices do not have determinants.

A determinant is a single real number associated with a square matrix only.

• Denoted by det ⁡A or ∣A∣ or Δ 

Order 1 (1×1 matrix):

∣A∣ = a

Order 2 (2×2 matrix):

∣A∣ = ad − bc

Order 3 (3×3 matrix):

\[A= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}\]

\[|A|=a_{11}(a_{22}a_{33}-a_{32}a_{23})-a_{12}(a_{21}a_{33}-a_{31}a_{23})+a_{13}(a_{21}a_{32}-a_{31}a_{22})\]

• If |A| = 0
  A matrix is called a Singular Matrix
• If |A| ≠ 0
  Matrix is called a Non-Singular Matrix