8.7. RANK OF A MATRIX 183

8.7 Rank of a MatrixDefinition 8.7.1 A submatrix of a matrix A is the rectangular array of numbers obtainedby deleting some rows and columns of A. Let A be an m×n matrix. The determinant rankof the matrix equals r where r is the largest number such that some r× r submatrix of Ahas a non zero determinant. The row rank is defined to be the dimension of the span of therows. The column rank is defined to be the dimension of the span of the columns.

Theorem 8.7.2 If A, an m×n matrix has determinant rank r, then there exist r rows of thematrix such that every other row is a linear combination of these r rows.

Proof: Suppose the determinant rank of A = (ai j) equals r. Thus some r× r subma-trix has non zero determinant and there is no larger square submatrix which has non zerodeterminant. Suppose such a submatrix is determined by the r columns whose indices arej1 < · · ·< jr and the r rows whose indices are i1 < · · ·< ir. I want to show that every rowis a linear combination of these rows. Consider the lth row and let p be an index between 1and n. Form the following (r+1)× (r+1) matrix

ai1 j1 · · · ai1 jr ai1 p...

......

air j1 · · · air jr air p

al j1 · · · al jr al p

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 r× r matrix inthe upper left corner. Thus

0 = al pC+r

∑k=1

Ckaik p

which implies

al p =r

∑k=1

−Ck

Caik p ≡

r

∑k=1

mkaik p

Since this is true for every p and since mk does not depend on p, this has shown the lth rowis a linear combination of the i1, i2, · · · , ir rows. ■

Corollary 8.7.3 The determinant rank equals the row rank.

Proof: From Theorem 8.7.2, every row is in the span of r rows where r is the determi-nant rank. Therefore, the row rank (dimension of the span of the rows) is no larger thanthe determinant rank. Could the row rank be smaller than the determinant rank? If so, it