lu decompositiongaussian eliminationadjugateminorcofactorrank conditionexistence conditionnon commutativitydeterminant relationtranspose inverseproduct inverseinverse of inverseuniquenessdeterminant conditionsingular matrixinvertible matrixidentity matrixdefinitioninverse matrixcomputationpropertiesfoundations

Definition

Given a square matrix of order $n$, the inverse of $A$, denoted $A^{- 1}$, is the matrix such that:

where $I$ is the identity matrix of order $n$. When such a matrix exists, it is unique. A square matrix that admits an inverse is called invertible or nonsingular. A matrix that does not admit an inverse is called singular.

The inverse matrix represents the linear transformation that reverses the effect of the original transformation. If $A$ maps a vector $\mathbf{x}$ to $\mathbf{b}$, that is $A \mathbf{x} = \mathbf{b}$, then $A^{- 1}$ maps $\mathbf{b}$ back to $\mathbf{x}$:

This is precisely the principle underlying the solution of systems of linear equations via the inverse matrix.

A square matrix $A$ is invertible if and only if its determinant is nonzero:

The condition $det ( A ) \neq 0$ is both necessary and sufficient for invertibility. It is equivalent to requiring that the rows (or columns) of $A$ are linearly independent, and that the rank of $A$ equals $n$. The set of all invertible matrices of order $n$ forms a group under matrix multiplication, known as the general linear group $G L ( n , \mathbb{R} )$, discussed in the entry on groups.

Properties of the inverse

The inverse matrix satisfies the following properties, for square matrices $A$ and $B$ of order $n$:

• $( A^{- 1} )^{- 1} = A$. The inverse of the inverse is the original matrix.
• $( A B )^{- 1} = B^{- 1} A^{- 1}$. The inverse of a product reverses the order of the factors.
• $( A^{T} )^{- 1} = ( A^{- 1} )^{T}$. The inverse of the transpose equals the transpose of the inverse.
• $det ( A^{- 1} ) = \frac{1}{det ( A )}$. The determinant of the inverse is the reciprocal of the determinant of $A$.

The reversal of order in $( A B )^{- 1} = B^{- 1} A^{- 1}$ is necessary for the same reason as in the transpose: matrix multiplication is not commutative, so inverting a product requires inverting each factor and reversing their order.

Computing the inverse: the cofactor method

The inverse of a square matrix $A$ of order $n$, when it exists, can be computed using the cofactor method. Given a square matrix $A = ( a_{i j} )$, the minor $M_{i j}$ is the determinant of the $( n - 1 ) \times ( n - 1 )$ submatrix obtained by deleting the $i$-th row and $j$-th column of $A$. The cofactor $C_{i j}$ is defined as:

The cofactor matrix $C$ is the matrix whose entry in position $( i , j )$ is the cofactor $C_{i j}$. The transpose of the cofactor matrix, denoted $C^{T}$, is called the adjugate of $A$ and written $adj ( A )$. The inverse is then given by:

The computation proceeds as follows: calculate the cofactor $C_{i j}$ for every entry of $A$, assemble the cofactor matrix $C$, take its transpose to obtain $adj ( A )$, and divide every entry by $det ( A )$.

Example

Consider the following matrix:

This is a lower triangular matrix. Its determinant is the product of the diagonal entries:

The cofactor matrix $C$ is assembled from the nine cofactors computed above:

Taking the transpose of $C$ gives the adjugate of $A$:

Dividing by $det ( A ) = 6$:

The cofactor method is exact but computationally expensive for large matrices, with complexity $O ( n ! )$ due to the determinant evaluations involved. In numerical practice, the inverse is typically computed via Gaussian elimination or LU decomposition, which achieve $O ( n^{3} )$ complexity.