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Use of determinants to solve systems of linear algebraic equations

1.2. Use of determinants to solve systems of linear algebraic equa-

tions

Let’s consider the system of three equations of the first order with three unknowns x 1 , x 2 , x 3 :


 a 11 x 1 + a 12 x 2 + a 13 x 3 = b 1 ,
a 21 x 1 + a 22 x 2 + a 23 x 3 = b 2 , (1.6)
a 31 x 1 + a 32 x 2 + a 33 x 3 = b 3 .

0
0
0
The triple of digits x , x , x is called the solution of system of linear equations (1.6), if those
1 2 3
digits were put instead of matching ones transformed all three equations into equality.
Let’s look at such determinants:

a 11 a 12 a 13 b 1 a 12 a 13 a 11 b 1 a 13 a 11 a 12 b 1

∆ = a 21 a 22 a 23 , ∆ 1 = b 2 a 22 a 23 , ∆ 2 = a 21 b 2 a 23 , ∆ 3 = a 21 a 22 b 2 .

a 31 a 32 a 33 b 3 a 32 a 33 a 31 b 3 a 33 a 31 a 32 b 3
The determinant ∆ is called the main determinant of the system (1.6). The determinants ∆ 1 ,
∆ 2 , ∆ 3 , received from the determinant by swapping matching columns into a column of free
elements are called auxiliary ones.
Let’s discuss such two cases: when the main determinant doesn’t equal to zero and when it
does equal to zero.
The first case: ∆ ̸= 0. Let’s prove that the solution of the system exists and it’s the only one.
To do this, let’s multiply both parts of the first equation of system (1.6) into cofactor A 11 , the
second into A 21 , the third into A 31 and then add these equations. We will get the result:
(a 11 · A 11 + a 21 · A 21 + a 31 · A 31 ) · x 1 + (a 12 · A 11 + a 22 · A 21 + a 32 · A 31 ) · x 2 +
+(a 13 · A 11 + a 23 · A 21 + a 33 · A 31 ) · x 3 = b 1 · A 11 + b 2 · A 21 + b 3 · A 31 .

Based on properties 9 and 10, we have: ∆·x 1 = b 1 ·A 11 +b 2 ·A 21 +b 3 ·A 31 = ∆ 1 . In the same way
we will get: ∆·x 2 = b 1 ·A 12 +b 2 ·A 22 +b 3 ·A 32 = ∆ 2 , ∆·x 3 = b 1 ·A 13 +b 2 ·A 23 +b 3 ·A 33 = ∆ 3 .
So:
∆ 1 ∆ 2 ∆ 3
x 1 = , x 2 = , x 3 = . (1.7)
∆ ∆ ∆
Formulas (1.7) are called Kramer’s formulas. To prove that the solution for system (1.6)
exists, it’s enough instead of x 1 , x 2 , x 3 put their meaning according to Kramer’s formulas and
make sure that three equations (1.6) are transformed into equality.
Kramer’s formulas also prove the only solution, since system (1.7) is the result of system
(1.6) and that’s why any solution of the system (1.6) is the solution of the system (1.7).
The first case. If the determinant of the system ∆ ̸= 0, then the only solution for this system
exists and can be calculated by Kramer’s formulas.


 x 1 + 2x 2 + x 3 = 4,
Example 1.3. Solve the system 3x 1 − 5x 2 + 3x 3 = 1,

2x 1 + 7x 2 − x 3 = 8.

Solution. As ∆ = 33 ̸= 0 then the given system has the only solution: x 1 = ∆ 1 = 33 = 1,
∆ 33
x 2 = ∆ 2 = 33 = 1, x 3 = ∆ 3 = 33 = 1. So, x 1 = 1, x 2 = 1, x 3 = 1 is the solution for this system.
∆ 33 ∆ 33

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