The inverse of a matrix should, like inverses of functions, operations, and numbers (arithmetic or multiplicative), satisfy . Here, a matrix inverse is a multiplicative operation, so like the reciprocal for real numbers, it must satisfy and . But what is this element, ?
Remember that matrices were originally created to solve . It makes some sense, then, that the identity matrix with respect to matrix multiplication is the diagonal matrix with all 's:
The inverse of a matrix is calculated by row reducing a related matrix, namely the matrix .
The inverse of a matrix is used in many contexts throughout linear algebra, including similar matrices, diagonalizable matrices, and almost any discussion of linear transformations involving matrices.
It is therefore helpful to know a little bit more about the inverse of an invertible matrix .
- The inverse of the inverse of a matrix is the matrix itself. That is, if , then .
- If , then and .
- The inverse commutes with transposition. That is, .
- Taking the inverse reverses the order of multiplication. That is,