3.3. THE MATHEMATICAL THEORY OF DETERMINANTS 99

By the formula for the expansion of a determinant along a column,

xi =1

det (A)det

∗ · · · y1 · · · ∗...

......

∗ · · · yn · · · ∗

 ,

where here the ith column of A is replaced with the column vector, (y1 · · · ·, yn)T , and thedeterminant of this modified matrix is taken and divided by det (A). This formula is knownas Cramer’s rule.

Definition 3.3.20 A matrix M , is upper triangular if Mij = 0 whenever i > j. Thus sucha matrix equals zero below the main diagonal, the entries of the form Mii as shown.

∗ ∗ · · · ∗

0 ∗. . .

......

. . .. . . ∗

0 · · · 0 ∗

A lower triangular matrix is defined similarly as a matrix for which all entries above themain diagonal are equal to zero.

With this definition, here is a simple corollary of Theorem 3.3.17.

Corollary 3.3.21 Let M be an upper (lower) triangular matrix. Then det (M) is obtainedby taking the product of the entries on the main diagonal.

3.3.7 Rank of a Matrix

Definition 3.3.22 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 determinantrank of the matrix equals r where r is the largest number such that some r × r submatrixof A has a non zero determinant. The row rank is defined to be the dimension of the spanof the rows. The column rank is defined to be the dimension of the span of the columns.

Theorem 3.3.23 If A, an m×n matrix has determinant rank r, then there exist r rows ofthe matrix such that every other row is a linear combination of these r rows.

Proof: Suppose the determinant rank of A having ijth entry aij equals r. Thus somer × r submatrix has non zero determinant and there is no larger square submatrix whichhas non zero determinant. Suppose such a submatrix is determined by the r columns whoseindices 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

ai1j1 · · · ai1jr ai1p...

......

airj1 · · · airjr airp

alj1 · · · aljr alp

