Note that this gives an easy way to write a formula for the inverse of an n×n matrix. Recall the definition of the inverse of a matrix in Definition 2.1.22 on Page 130.
ij the ijth cofactor of A.
−1 −1 aij = cof (A )jidet(A) .
The next corollary points out that if an n×n matrix A has a right or a left inverse, then it has an inverse.
Corollary 3.3.19 Let A be an n × n matrix and suppose there exists an n × n matrix B such that BA = I. Then A−1 exists and A−1 = B. Also, if there exists C an n × n matrix such that AC = I, then A−1 exists and A−1 = C.
( ) A −1 = (BA )A− 1 = B AA −1 = BI = B.
The conclusion of this corollary is that left inverses, right inverses and inverses are all the same in the context of n × n matrices.
Theorem 3.3.18 says that to find the inverse, take the transpose of the cofactor matrix and divide by the determinant. The transpose of the cofactor matrix is called the adjugate or sometimes the classical adjoint of the matrix A. It is an abomination to call it the adjoint although you do sometimes see it referred to in this way. In words, A−1 is equal to one over the determinant of A times the adjugate matrix of A.
. This formula is known as Cramer’s rule.
Definition 3.3.20 A matrix M, is upper triangular if Mij = 0 whenever i > j. Thus such a matrix equals zero below the main diagonal, the entries of the form Mii as shown.
A lower triangular matrix is defined similarly as a matrix for which all entries above the main diagonal are equal to zero.
With this definition, here is a simple corollary of Theorem 3.3.17.
is obtained by taking the product of the entries on the main diagonal.