Editing Invertible matrix (section)

=== Other properties ===
Furthermore, the following properties hold for an invertible matrix {{math|'''A'''}}:

* <math>(\mathbf A^{-1})^{-1} = \mathbf A</math>
* <math>(k \mathbf A)^{-1} = k^{-1} \mathbf A^{-1}</math> for nonzero scalar {{mvar|k}}
* <math>(\mathbf{Ax})^+ = \mathbf x^+ \mathbf A^{-1}</math> if {{math|'''A'''}} has orthonormal columns, where {{math|{{sup|+}}}} denotes the [[Moore–Penrose inverse]] and {{math|'''x'''}} is a vector
* <math>(\mathbf A^\mathrm{T})^{-1} = (\mathbf A^{-1})^\mathrm{T}</math>
* For any invertible {{mvar|n}}-by-{{mvar|n}} matrices {{math|'''A'''}} and {{math|'''B'''}}, <math>(\mathbf{AB})^{-1} = \mathbf B^{-1} \mathbf A^{-1}.</math> More generally, if <math>\mathbf A_1, \dots, \mathbf A_k</math> are invertible {{mvar|n}}-by-{{mvar|n}} matrices, then <math>(\mathbf A_1 \mathbf A_2 \cdots \mathbf A_{k-1} \mathbf A_k)^{-1} = \mathbf A_k^{-1} \mathbf A_{k-1}^{-1} \cdots \mathbf A_2^{-1} \mathbf A_1^{-1}.</math>
*<math>\det \mathbf A^{-1} = (\det \mathbf A)^{-1}.</math>

The rows of the inverse matrix {{math|'''V'''}} of a matrix {{math|'''U'''}} are [[orthonormal]] to the columns of {{math|'''U'''}} (and vice versa interchanging rows for columns). To see this, suppose that {{math|1='''UV''' = '''VU''' = '''I'''}} where the rows of {{math|'''V'''}} are denoted as <math>v_i^{\mathrm{T}}</math> and the columns of {{math|'''U'''}} as <math>u_j</math> for <math>1 \leq i,j \leq n.</math> Then clearly, the [[Dot product|Euclidean inner product]] of any two <math>v_i^{\mathrm{T}} u_j = \delta_{i,j}.</math> This property can also be useful in constructing the inverse of a square matrix in some instances, where a set of [[orthogonal]] vectors (but not necessarily orthonormal vectors) to the columns of {{math|'''U'''}} are known. In which case, one can apply the iterative [[Gram–Schmidt process]] to this initial set to determine the rows of the inverse {{math|'''V'''}}. 

A matrix that is its own inverse (i.e., a matrix {{math|'''A'''}} such that {{math|1='''A''' = '''A'''{{sup|−1}}}} and consequently {{math|1='''A'''{{sup|2}} = '''I'''}}) is called an [[involutory matrix]].