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Elements of Determinants’ Theory

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Example 1.1. ∆ = = 2 · (−4) − 1 · 3 = −11. ,
3 −4
In an analogy, the third order determinant is introduced, and it also corresponds to the square
table, which consists of nine elements. The determinant is calculated by the following rule:

a 11 a 12 a 13

a 21 a 22 a 23 = a 11 ·a 22 ·a 33 +a 12 a 23 a 31 +a 21 a 32 a 13 −a 13 a 22 a 31 −a 23 a 32 a 11 −a 12 a 21 a 33 (1.3)

a 31 a 32 a 33
The diagonal created by elements a 11 , a 22 , a 33 is called the main one, and the diagonal created
by elements a 13 , a 22 , a 31 — the side one.

3 −2 1

Example 1.2. −2 1 3 = 3 · 1 · (−2) + (−2) · 3 · 2 + (−2) · 0 · 1 − 1 · 1 · 2 − 3 ·

2 0 −2
0 · 3 − (−2) · (−2) · (−2) = −12. ,
Properties of determinants.
1. Determinant doesn’t change if its rows and columns have switched.:

a 11 a 12 a 13 a 11 a 21 a 31

a 21 a 22 a 23 = a 12 a 22 a 32 .

a 31 a 32 a 33 a 13 a 23 a 33
This operation is called transposition.
2. On changing two rows or columns of a determinant, it will turn it’s sign
into the opposite one.

a 11 a 12 a 13 a 12 a 11 a 13

For example: a 21 a 22 a 23 = − a 22 a 21 a 23 .

a 31 a 32 a 33 a 32 a 31 a 33
3. If determinant has two similar rows or columns than it equals to zero.
PROOF. When changing two similar rows or columns of a determinant, it won’t change,
but according to property 2 its sign will change. So, ∆ = −∆, or ∆ = 0.
4. Multiplying of all elements of some row or column by some digit k equals
to multiplying the determinant by digit k.

a 11 a 12 k · a 13 a 11 a 12 a 13

For example: a 21 a 22 k · a 23 = k · a 21 a 22 a 23 .

a 31 a 32 k · a 33 a 31 a 32 a 33
Remark 1.1. Common multiple of all elements of some column or a row can
be carried away from the determinant.
5. If all elements of some column or row are equal to zero, than the deter-
minant equals to zero.
This property comes out from property 4 (when k = 0).
6. If elements of two columns or rows of the determinant are proportional
than the determinant equals to zero.
Indeed, if elements of two rows are proportional than according to property 4 common
multiple can be carried away from the determinant. As a result, the determinant with two
equal rows will be left, which equals to zero according to property 3.

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