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umn from the second matrix. Of course, all three results should give the same
answer, which they do.

8.5 Inverse of a Matrix

In this section, we assume that we are dealing with square matrices (n ⊗ n)
because these are the only class of matrices for which we can define an
inverse.

–1
DEFINITION A matrix M is called the inverse of matrix M if the following
conditions are satisfied:

−1
MM −1 = M M = I (8.4)

(The identity matrix is the (n ⊗ n) matrix with ones on the diagonal and zero
everywhere else; the matrix eye(n,n)in MATLAB.)

EXISTENCE The existence of an inverse of a matrix hinges on the condition
that the determinant of this matrix is non-zero [det(M) in MATLAB]. We leave
the proof of this theorem to future courses in linear algebra. For now, the for-
mula for generating the value of the determinant is given here.

• The determinant of a square matrix M, of size (n ⊗ n), is a number
equal to:

∑ P
det(M = (−1 MM 2 2 k M 3 3 k …M nk n (8.5)
)
)
1 1 k
P
where P is the n! permutation of the first n-integers. The sign in front of each
term is positive if the number of transpositions relating

( 12 3, , ,… ) n and 1 (k k k 3 ,… )
,
,
, k
,
2
n
is even, while the sign is negative otherwise.
Example 8.2
Using the definition for a determinant, as given in Eq. (8.5), find the expres-
sion for the determinant of a (2 ⊗ 2) and a (3 ⊗ 3) matrix.

© 2001 by CRC Press LLC