The determinant of Ais then obtained by multiplying all the elements on the main
diagonal and
|A|= det(A) = a 11 a 22... ann =
∏n
i=1
aii.
Rank of a Matrix
The rank of a m×nmatrix Ais denoted using the notation rank (A). The rank of
the matrix Ais a real number defined as the size of the largest nonzero determinant
that can be formed using the elements of A. If A= (aij)m×n, one can show that the
maximum possible rank (A)is the smaller of the numbers mand n.
Calculation of the Inverse Matrix
The following illustrates some methods for calculating the inverse of a square
matrix if such an inverse exists. Previously it has been shown that if Cis the cofactor
matrix of A, then
AC T=|A|I. (10 .21)
By multiplying this equation on the left by A−^1 and dividing by |A|,one can verify
the result
A−^1 =^1
|A|
CT. (10 .22)
as a formula for calculating the inverse matrix. Define the transpose of the cofactor
matrix CT to be the adjoint of A. The notation AdjAis used to denote the adjoint
matrix. Using this definition, the above results can be expressed in the form
(AdjA)A=A(AdjA) = |A|I or A−^1 =^1
|A|
Adj A (10 .23)
If Ais an n×nsquare matrix and the determinant satisfies det A=|A|= 0,then
Ais called a singular matrix. If Ais singular, then the inverse matrix does not exist.
If det A=|A|= 0, then Ais called a nonsingular matrix , and the inverse matrix A−^1
exists under these conditions as can be discerned by examining the equation (10.23).
Example 10-21.
Find the inverse of the matrix A=
[
1 2
−3 4
]
.
Solution: The cofactor matrix associated with Ais given by C=
[
4 3
−2 1
]
and
