534 CHAPTER 28. THE MATHEMATICAL THEORY OF DETERMINANTS∗

Now considern

∑i=1

air cof(A)ik det(A)−1

when k ̸= r. Replace the kth column with the rth column to obtain a matrix Bk whosedeterminant equals zero by Corollary 28.1.6. However, expanding this matrix along the kth

column yields

0 = det(Bk)det(A)−1 =n

∑i=1

air cof(A)ik det(A)−1

Summarizing,n

∑i=1

air cof(A)ik det(A)−1 = δ rk.

Using the other formula in Theorem 28.1.13, and similar reasoning,

n

∑j=1

ar j cof(A)k j det(A)−1 = δ rk

This proves that if det(A) ̸= 0, then A−1 exists with A−1 =(

a−1i j

), where

a−1i j = cof(A) ji det(A)−1 .

Now suppose A−1 exists. Then by Theorem 28.1.10,

1 = det(I) = det(AA−1)= det(A)det

(A−1)

so det(A) ̸= 0. ■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 28.1.15 Let A be an n× n matrix and suppose there exists an n× n matrix Bsuch that BA = I. Then A−1 exists and A−1 = B. Also, if there exists C an n×n matrix suchthat AC = I, then A−1 exists and A−1 =C.

Proof: Since BA = I, Theorem 28.1.10 implies

detBdetA = 1

and so detA ̸= 0. Therefore from Theorem 28.1.14, A−1 exists. Therefore,

A−1 = (BA)A−1 = B(AA−1)= BI = B.

The case where CA = I is handled similarly. ■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 28.1.14 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 adjugateor sometimes the classical adjoint of the matrix A. It is an abomination to call it the adjointalthough you do sometimes see it referred to in this way. In words, A−1 is equal to one overthe determinant of A times the adjugate matrix of A.