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5.5. SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS 203

5.5.2-4. General system of m linear equations with n unknown quantities.

Suppose that system (5.5.1.1) is consistent and its basic matrix A has rank r.First, in
the matrix A, one finds a submatrix of size r × r with nonzero rth-order determinant
and drops the m – r equations whose coefficients do not belong to this submatrix (the
dropped equations follow from the remaining ones and can, therefore, be neglected). In
the remaining equations, the n – r unknown quantities (free unknown quantities) that are
not involved in the said submatrix should be transferred to the right-hand sides. Thus, one
obtains a system of r equations with r unknown quantities, which can be solved by any of
the methods described in Paragraph 5.5.2-2.
Remark. If the rank r of the basic matrix and the rank of the augmented matrix of system (5.5.1.1) are
equal to the number of the unknown quantities n, then the system has a unique solution.

5.5.2-5. Solutions of homogeneous and corresponding nonhomogeneous systems.

1 . Suppose that the basic matrix A of the homogeneous system (5.5.1.3) has rank r and
◦
its submatrix in the left top corner, B =[a ij ](i, j = 1, ... , r), is nondegenerate. Let
M =det B ≠ 0 be the determinant of that submatrix. Any solution x 1 , ... , x n has n – r
free components x r+1 , ... , x n and its first components x 1 , ... , x r are expressed via the free
components as follows:
1
x 1 =– [x r+1 M 1 (a i(r+1) )+ x r+2 M 1 (a i(r+2) )+ ··· + x n M 1 (a in )],
M
1
x 2 =– [x r+1 M 2 (a i(r+1) )+ x r+2 M 2 (a i(r+2) )+ ··· + x n M 2 (a in )], (5.5.2.4)
M
...... ......... ........ ........ ........ ........ ......... ......
1
x r =– [x r+1 M r (a i(r+1) )+ x r+2 M r (a i(r+2) )+ ··· + x n M r (a in )],
M
where M j (a ik ) is the determinant of the matrix obtained from B by replacing its jth column
with the column whose components are a 1k , a 2k , ... , a rk :

a 11 a 12 ... a 1k ... a 1r

a 21 a 22 ... a 2k ... a 2r
M j (a ik )= . . . . . . .
. . . . .
. . . . . .
.
a r1 a r2 a rk a rr
... ...
◦
2 . Using (5.5.2.4), we obtain the following n – r linearly independent solutions of the
original system (5.5.1.3):

M 1 (a i(r+1) ) M 2 (a i(r+1) ) M r (a i(r+1) )
X 1 = – – ··· – 1 0 ··· 0 ,
M M M

M 1 (a i(r+2) ) M 2 (a i(r+2) ) M r (a i(r+2) )
X 2 = – – ··· – 0 1 ··· 0 ,
M M M

M 1 (a in ) M 2 (a in ) M r (a in )
X n–r = – – ··· – 0 0 ··· 1 .
M M M
Any solution of system (5.5.1.3) can be represented as their linear combination
X = C 1 X 1 + C 2 X 2 + ··· + C n–r X n–r , (5.5.2.5)
where C 1 , C 2 , ... , C n–r are arbitrary constants. This formula gives the general solution of
the homogeneous system.