Recently I was playing around with matrices and noticed that matrices such as and are equal to their inverse. In other words, they are matrices such that or where is the 2 by 2 identity matrix. This led me to wonder which 2×2 matrices in general had this property. I may have worked this out previously but I could not recall the condition. In this post we’ll see the general form of such matrices and then generalise the result to higher dimensions.
For 2×2 matrices the inverse of the matrix is , where .
Hence matching one of the entries of these two matrices, . From this, either or .
1) If , the inverse becomes
from which and (and hence or ). This leads to the forms .
2) If , the inverse becomes
from which . This leads to the form where .
Combining the two cases (and noting that there is overlap between the two) , we conclude that the 2×2 matrices that are inverses of themselves are either
- where .
To check for this final case we simply need to ensure that (1) the matrix has determinant -1 and (2) the matrix has diagonal entries summing to 0. For example, the matrix has this property so we can immediately say , or indeed if we desire. 🙂
An alternative approach, that works more generally for nxn matrices, uses some knowledge of linear algebra. Our desired matrices satisfy the equation or . Hence the minimal polynomial of is a factor of . If it is either or this corresponds to the solutions . If the minimal polynomial is , the fact that there are no repeated factors (i.e. each root has multiplicity 1) implies that is diagonalisable. In this case, is similar to a diagonal matrix with +1 or -1 as the diagonal entries.
Hence or where , and represents the identity matrix (and is an submatrix of 0s). In the special case of this means we can also write with and and obtain
Note here that the diagonal entries sum to 0 and .
Matrices that satisfy or are known as involutory. Such matrices have applications in cryptography where it may be useful for the same matrix operation to act as its inverse in decryption. We have the following two further characterisations of involutory matrices (holding true in any field of characteristic not equal to 2).
- , where satisfies ( is then known as an idempotent matrix).
To see this, simply note that if , the matrix is idempotent since . Conversely, if , then .
- (See ) or where is , is and .
To see this, we write as before and partition and as , where is and is . Then while
where and .
The converse (that if has this form then it is involutory) is easier to prove: if then
As an example, since , the matrix
has the property that or .
 J. Levine and H. M. Nahikian, On the Construction of Involutory Matrices, The American Mathematical Monthly, Vol. 69, No. 4 (Apr 1962), pp. 267-272.