20.2. THE DETERMINANT 439

20.2.3, when two rows or two columns in a matrix M, are switched, this results in multi-plying the determinant of the old matrix by −1 to get the determinant of the new matrix.Therefore, by Lemma 20.2.11,

det(B j) = (−1)n− j (−1)n−i det((

Ai j ∗0 ai j

))= (−1)i+ j det

((Ai j ∗0 ai j

))= ai j cof(A)i j .

Therefore,

det(A) =n

∑j=1

ai j cof(A)i j

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.

20.2.8 Row, Column, and Determinant RankThis section will consider the concept of rank of a matrix. This is a number and its descrip-tion is in the following definition.

Definition 20.2.14 A sub-matrix of a matrix A is the rectangular array of num-bers obtained by deleting some rows and columns of A. Let A be an m× n matrix. Thedeterminant rank of the matrix equals r where r is the largest number such that some r× rsub-matrix of A has a non zero determinant. The row rank is defined to be the dimensionof the span of the rows. The column rank is defined to be the dimension of the span of thecolumns.

Theorem 20.2.15 If A, an m×n matrix has determinant rank, r, then there exist rrows of the matrix 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 are

j1 < · · ·< jr

and the r rows whose indices arei1 < · · ·< ir

I want to show that every row is a linear combination of these rows. Consider the lth rowand let p be an index between 1 and n. Form the following (r+1)× (r+1) matrix

ai1 j1 · · · ai1 jr ai1 p...

......

air j1 · · · air jr air pal j1 · · · al jr al p

