Linear Algebra, Theory And Applications, 2012a

94 DETERMINANTS
By the formula for the expansion of a determinant along a column,
⎛
⎞
∗ ··· y 1 ··· ∗
1
x i =
det (A) det ⎜
⎟
⎝ . . . ⎠ ,
∗ ··· y n ··· ∗
where here the i th column of A is replaced with the column vector, (y 1 ····,y n ) T ,andthe
determinant of this modified matrix is taken and divided by det (A). This formula is known
as Cramer’s rule.
Definition 3.3.20 A matrix M, is upper triangular if M ij =0whenever i>j.Thussuch
a matrix equals zero below the main diagonal, the entries of the form M ii as shown.
⎛
⎞
∗ ∗ ··· ∗
. 0 ∗ ..
. ⎜
⎝
.
. .. . ⎟ ..
∗ ⎠
0 ··· 0 ∗
A lower triangular matrix is defined similarly as a matrix for which all entries above the
main 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 obtained
by 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 obtained
by deleting some rows and columns of A. Let A be an m × n matrix. The determinant
rank of the matrix equals r where r is the largest number such that some r × r submatrix
of A has a non zero determinant. The row rank is defined to be the dimension of the span
of 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 of
the matrix such that every other row is a linear combination of these r rows.
Proof: Suppose the determinant rank of A =(a ij ) equals r. Thus some r × r submatrix
has non zero determinant and there is no larger square submatrix which has non zero
determinant. Suppose such a submatrix is determined by the r columns whose indices are
and the r rows whose indices are
j 1 < ···