Mathematical Methods for Physicists: A concise introduction - Site Map

SYSTEMS OF LINEAR EQUATIONS
The a 0
jk are required to construct ~ A 1 . Since ~ A ~ A 1 ˆ ~I, we have
a 11 a 0 11 ‡ a 12 a 0 12 ‡‡a 1n a 0 1n ˆ 1;
a 21 a21 0 ‡ a 22 a22 0 ‡‡a 2n a2n 0 ˆ 0;
.
a n1 a 0 n1 ‡ a n2 a 0 n2 ‡‡a nn a 0 nn ˆ 0:
…3:22†
The solution to the above set of linear algebraic equations (3.22) may be facilitated
by applying Cramer's rule. Thus
ajk 0 ˆ cofactor a kj
det A ~ : …3:23†
From (3.23) it is clear that ~A 1 exists if and only if matrix ~A is non-singular (that
is, det ~A 6ˆ 0).
Systems of linear equations and the inverse of a matrix
As an immediate application, let us apply the concept of an inverse matrix to a
system of n linear equations in n unknowns …x 1 ; ...; x n †:
in matrix form we have
a 11 x 1 ‡ a 12 x 2 ‡‡a 1n x n ˆ b 1 ;
a 21 x 2 ‡ a 22 x 2 ‡‡a 2n x n ˆ b 2 ;
.
.
a n1 x n ‡ a n2 x n ‡‡a nn x n ˆ b n ;
~A ~X ˆ ~B; …3:24†
where
0
1
a 11 a 12 ... a 1n
a 21 a 22 ... a 2n
~A ˆ
B . . .
C
@ . . . A ;
0
x 1
1
x 2
~X ˆ
B .
C
@ . A ;
0
b 1
1
b 2
~B ˆ
B .
C
@ . A :
a n1 a n2 ... a nn
We can prove that the above linear system possesses a unique solution given by
x n
~X ˆ ~A 1 ~B: …3:25†
The proof is simple. If ~A is non-singular it has a unique inverse ~A 1 . Now premultiplying
(3.24) by ~A 1 we obtain
~A 1 … ~A ~X† ˆ ~A 1 ~B;
b n
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