48 CHAPTER 1. REVIEW OF SOME LINEAR ALGEBRA

which is the formula for expanding det(A) along the ith row. Also,

det(A) = det(AT )= n

∑j=1

aTi j cof

(AT )

i j =n

∑j=1

a ji cof(A) ji

which is the formula for expanding det(A) along the ith column. ■

1.9.6 A Formula for the InverseNote that this gives an easy way to write a formula for the inverse of an n×n matrix. Recallthe definition of the inverse of a matrix in Definition 1.6.7 on Page 29.

Theorem 1.9.18 A−1 exists if and only if det(A) ̸= 0. If det(A) ̸= 0, then A−1 =(a−1

i j

)where

a−1i j = det(A)−1 cof(A) ji

for cof(A)i j the i jth cofactor of A.

Proof: By Theorem 1.9.17 and letting (air) = A, if det(A) ̸= 0,

n

∑i=1

air cof(A)ir det(A)−1 = det(A)det(A)−1 = 1.

Now in the matrix A, replace the kth column with the rth column and then expand along thekth column. This yields for k ̸= r,

n

∑i=1

air cof(A)ik det(A)−1 = 0

because there are two equal columns by Corollary 1.9.9. Summarizing,

n

∑i=1

air cof(A)ik det(A)−1 = δ rk.

Using the other formula in Theorem 1.9.17, 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 1.9.13,

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.