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5.5. SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS 199

5.5.2-2. General square system of linear equations.

A square system of linear equations has the form
AX = B, (5.5.2.2)
where A is a square matrix.

◦
1 . If the determinant of system (5.5.2.2) is different from zero, i.e. det A ≠ 0, then the
system has a unique solution,
–1
X = A B.
2 . Cramer rule. If the determinant of the matrix of system (5.5.2.2) is different from zero,
◦
i.e. Δ =det A ≠ 0, then the system admits a unique solution, which is expressed by
Δ 1 Δ 2 Δ n
x 1 = , x 2 = , ... , x n = , (5.5.2.3)
Δ Δ Δ
where Δ k (k = 1, 2, ... , n) is the determinant of the matrix obtained from A by replacing
its kth column with the column of free terms:

a 11 a 12 ... b 1 ... a 1n

a 21 a 22 ... b 2 ... a 2n
Δ k = . . . . . . .
. . . . .
. . . . . .
.
a n1 a n2 ... b n ... a nn

Example 1. Using the Cramer rule, let us find the solution of the system of linear equations
2x 1 + x 2 + 4x 3 = 16,
3x 1 + 2x 2 + x 3 = 10,
x 1 + 3x 2 + 3x 3 = 16.
The determinant of its basic matrix is different from zero,

214
Δ = 321 = 26 ≠ 0,

133

and we have
216 21

1614 4 16
Δ 1 = 1021 = 26, Δ 2 = 310 1 = 52, Δ 3 = 32 10 = 78.

1633 116 3 13 16

Therefore, by the Cramer rule (5.5.2.3), the only solution of the system has the form
Δ 1 26 Δ 2 52 Δ 3 78
x 1 = = = 1, x 2 = = = 2, x 3 = = = 3.
Δ 26 Δ 26 Δ 26
◦
3 . Gaussian elimination of unknown quantities.
Two systems are said to be equivalent if their sets of solutions coincide.
The method of Gaussian elimination consists in the reduction of a given system to an
equivalent system with an upper triangular basic matrix. The latter system can be easily
solved. This reduction is carried out in finitely many steps. On every step, one performs an
elementary transformation of the system (or the corresponding augmented matrix) and ob-
tains an equivalent system. The elementary transformations are of the following three types:
1. Interchange of two equations (or the corresponding rows of the augmented matrix).
2. Multiplication of both sides of one equation (or the corresponding row of the augmented
matrix) by a nonzero constant.
3. Adding to both sides of one equation both sides of another equation multiplied by a
nonzeroconstant (adding to some row of the augmented matrixits another row multiplied
by a nonzero constant).