Here are some classes of real non-singular matrices that are easier to invert than most. In an earlier post I had referred to involutory matrices that had the special property that they were equal to their inverse – hence they are the easiest types of matrix to invert!

- 1 by 1 and 2 by 2 matrices are easy to invert:

- Diagonal matrices (including the identity matrix) are among the easiest to invert:

- Orthogonal matrices satisfy . Special cases are permutation matrices and rotation matrices.

- A matrix with all 1s down its diagonal and non-zero entries in a single row or column are easily invertible (atomic triangular matrices are a special case):

Other special cases of this are elementary matrices corresponding to the addition of a row to another:

- Bidiagonal matrices with 1s down the main diagonal are inverted as follows:

A special case of this is that the sum matrix and difference matrix are inverses:

Another special case is the following alternating matrix:

If one thinks about matrix inversion in terms of Gauss-Jordan elimination, keeping track of the order in which row/column operations can be done allow us to carry out matrix inversions such as the following:

- Integer-valued matrices that are equivalent (up to row or column operations which don’t change the determinant) to a triangle matrix with 1 or -1 as diagonal entries have inverses that are integer-valued. See [1] for more details.

- Matrices that are a sum of the identity matrix and a matrix with all entries equal to a constant are inverted as follows.

If , then This is a special case of the formula . For column vectors and this becomes .

- The inverse of the second difference matrix is

- Using identities such as , , and , where and are easily invertible, leads to simplifications. For example, if is diagonal, multiplies the columns of by the entries of and so will multiply the rows of by the entries of .

- The identity (where is the Kronecker product) leads to identities such as

- The following block matrix results also may be useful [2].

#### References

[1] R. Hanson, Integer Matrices Whose Inverse Contain Only Integers, The Two-Year College Mathematics Journal, Vol. 13, No. 1 (Jan., 1982), pp. 18-21.

[2] D. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas, Princeton University Press, 2011.

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