Invertible matrix

A square matrix is said to be invertible if there exists another square matrix that, when multiplied with it, yields the Identity matrix. In symbols, A\mathrm{A} is invertible n×nn\times n matrix if there exists another n×nn\times n matrix A−1\mathrm{A}^{-1} such that

AA−1=A−1A=In\mathrm{A}\mathrm{A}^{-1}=\mathrm{A}^{-1}\mathrm{A}=\mathrm{I}_{n}

A−1\mathrm{A}^{-1} is known as the inverse of A\mathrm{A}.

Characterization

There are a lot of ways to determine if a matrix is invertible. The following are a few and all equivalent to each other and to the definition above:

  • A\mathrm{A} has a full rank A\mathrm{A}.
  • The transpose of A\mathrm{A}, AT\mathrm{A}^{T}, is invertible.
  • The determinant of A\mathrm{A} is nonzero, det⁡A≠0\det \mathrm{A}\neq0.
  • All of the columns of A\mathrm{A} are linearly independent.
  • All of the rows of A\mathrm{A} are linearly independent.
  • The number 00 is not an eigenvalue of A\mathrm{A}.

Properties

  • (A−1)−1=A(\mathrm{A}^{-1})^{-1}=\mathrm{A}
  • (cA)−1=c−1A−1(c\mathrm{A})^{-1}=c^{-1}\mathrm{A}^{-1}
  • (AT)−1=(A−1)T(\mathrm{A}^{T})^{-1}=(\mathrm{A}^{-1})^{T}
  • det⁡(A−1)=(det⁡A)−1\det(\mathrm{A}^{-1})=(\det \mathrm{A})^{-1}
  • If A\mathrm{A} and B\mathrm{B} are both invertible n×nn\times n matrices, (AB)−1=A−1B−1(\mathrm{A}\mathrm{B})^{-1}=\mathrm{A}^{-1}\mathrm{B}^{-1}

Diagonalization

If A\mathrm{A} can be eigendecomposed into A=SΛS−1\mathrm{A}=\mathrm{S}\Lambda \mathrm{S}^{-1} then the inverse can be found by inverting the eigenvalue matrix only:

A−1=SΛ−1S−1\mathrm{A}^{-1}=\mathrm{S}\Lambda^{-1}\mathrm{S}^{-1}