440 CHAPTER 20. THEORY OF DETERMINANTS∗

Of course you can assume l /∈ {i1, · · · , ir} because there is nothing to prove if the lth rowis one of the chosen ones. The above matrix has determinant 0. This is because if p /∈{ j1, · · · , jr} then the above would be a submatrix of A which is too large to have non zerodeterminant. On the other hand, if p ∈ { j1, · · · , jr} then the above matrix has two columnswhich are equal so its determinant is still 0.

Expand the determinant of the above matrix along the last column. Let Ck denote thecofactor associated with the entry aik p. This is not dependent on the choice of p. Remember,you delete the column and the row the entry is in and take the determinant of what is leftand multiply by −1 raised to an appropriate power. Let C denote the cofactor associatedwith al p. This is given to be nonzero, it being the determinant of the matrix ai1 j1 · · · ai1 jr

......

air j1 · · · air jr

Thus 0 = al pC+∑

rk=1 Ckaik p which implies al p = ∑

rk=1

−CkC aik p ≡ ∑

rk=1 mkaik p. Since this

is true for every p and since mk does not depend on p, this has shown the lth row is a linearcombination of the i1, i2, · · · , ir rows.

Corollary 20.2.16 The determinant rank equals the row rank.

Proof: From Theorem 20.2.15, the row rank is no larger than the determinant rank.Could the row rank be smaller than the determinant rank? If so, there exist p rows for p < rsuch that the span of these p rows equals the row space. But this implies that the r× r sub-matrix whose determinant is nonzero also has row rank no larger than p which is impossibleif its determinant is to be nonzero because at least one row is a linear combination of theothers.

Corollary 20.2.17 If A has determinant rank r, then there exist r columns of the matrixsuch that every other column is a linear combination of these r columns. Also the columnrank equals the determinant rank.

Proof: This follows from the above by considering AT . The rows of AT are the columnsof A and the determinant rank of AT and A are the same. Therefore, from Corollary 20.2.16,column rank of A = row rank of AT = determinant rank of AT = determinant rank of A.

20.2.9 Formula for the InverseNote that this gives an easy way to write a formula for the inverse of an n×n matrix.

Theorem 20.2.18 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 20.2.13 and letting (air) = A, if det(A) ̸= 0,

n

∑i=1

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