Page 249 - Advanced engineering mathematics

7.7 Matrix Inverses 229

Conclusion (8) follows from (7).
Conclusion (9) follows immediately from the discussion preceding Theorem 7.6.
Finally, for conclusion (10), first suppose AX = B has a solution for every n × 1matrix B.
Let X j be the solution of
⎛ ⎞
0
⎜ 0 ⎟
⎜ ⎟
⎜.⎟
.
⎜ . ⎟
⎜ ⎟
AX = 1
⎜ ⎟
⎜ ⎟
0
⎜ ⎟
⎜ ⎟
⎜ .⎟
.
⎝ . ⎠
0
with 1 in row j and all other elements zero. Then X 1 ,··· ,X n form the columns of an n × n
matrix K and it is routine to check that AK = I n , hence K = A −1 and A is nonsingular.
−1
Conversely, if A is nonsingular, then X = A B is the solution of AX = B for any n × 1
matrix B.
Matrix inverses relate to systems of linear equations in the following way.
THEOREM 7.16

Let A be n × n.
1. A homogeneous system AX = O has a nontrivial solution if and only if A is singular.
2. A consistent nonhomogeneous system AX = B has a unique solution if and only if A is
nonsingular. In this case the solution is
X = A B.
−1
Proof If A is singular, then A R = I n by Theorem 7.15, conclusion (5), so the system AX = O
has a nontrivial solution by Corollary 7.3.
Conversely, suppose the system AX = O has a nontrivial solution. Then rank(A)< n by
Theorem 7.15, conclusion (6), so A is singular.
This proves conclusion (1). For conclusion (2), suppose the system is consistent. The general
solution has the form X=H+U p , where H is the general solution of the associated homogeneous
system. Therefore the given system has a unique solution exactly when the homogeneous system
has only the trivial solution, which occurs if and only if A is nonsingular.

Finding the inverse of a nonsingular matrix is most easily done using a software routine. In
the linalg package of linear algebra routines of MAPLE, the inverse of a matrix A that has
been entered can be found using

inverse(A);

If it happens that A is singular, the routine will return this conclusion.
Despite this, it is sometimes useful to understand a procedure for finding a matrix inverse.
.
.
Let A be an n × n matrix. Form the n × 2n matrix [I n .A] whose first n columns are A and whose
second n columns are I n . For example, if

2 3
A =
−19

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October 14, 2010 14:23 THM/NEIL Page-229 27410_07_ch07_p187-246