APP. A] VECTORS AND MATRICES 421
Part A (Reduction): Reduce the augmented matrixMto echelon form. If a row of the form( 0 , 0 ,..., 0 ,b), with
b=0, appears, thenstop. The system does not have a solution.
Part B (Back-Substitution): Further reduce the augmented matrixMto its row canonical form.
The unique solution of the system or, when the solution is not unique, the free variable form of the solution
is easily obtained from the row canonical form ofM.
The following example applies the above algorithm to a systemSwith a unique solution. The cases whereS
has no solution and whereShas an infinite number of solutions are shown in Problem A.23.
EXAMPLE A.11
Solve the system:
⎧
⎨
⎩x+ 2 y+z= 3
2 x+ 5 y−z=− 4
3 x− 2 y−z= 5
Reduce its augmented matrixMto echelon form and then to row canonical form as follows:M=⎡
⎣1213
25 − 1 − 4
3 − 2 − 15⎤
⎦∼⎡
⎣121 3
01 − 3 − 10
0 − 8 − 4 − 4⎤
⎦∼⎡
⎣1213
01 − 3 − 10
00 − 28 − 84⎤
⎦∼⎡
⎣12 1 3
01 − 3 − 10
00 1 3⎤
⎦∼⎡
⎣120 0
010 − 1
001 3⎤
⎦∼⎡
⎣100 2
010 − 1
001 3⎤
⎦Thus the system has the unique solutionx=2,y=−1,z=3 or, equivalently, the vectoru=( 2 ,− 1 , 3 ).We
note that the echelon form ofMalready indicated that the solution was unique since it corresponded to a triangular
system.
Inverse of ann×nMatrix
Figure A-7 contains Algorithm A-3 which finds the inverseA−^1 of any arbitraryn×nmatrix.Fig. A-7