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Elements of Determinants’ Theory

Calculating of the determinant by any formula (1.5) is called the expansion of it by elements
of some row or column.
Property 9 gives another rule of calculating the determinant of the third order. It’s better to

3 −2 1

calculate the determinant −2 1 3 by elements of the third row or second column

2 0 −2
as there is zero element there. Then, for example

−2 1 3 −2
∆ = a 31 · A 31 + a 32 · A 32 + a 33 · A 33 = 2 · + (−2) · =
1 3 −2 1
= 2 · (−7) + (−2) · (−1) = −14 + 2 = −12.

The calculation of this determinant will be more rational if property 8 is used with it, which
will give the opportunity to get the additional zero element in third row. To do this it’s
enough, for example, to add elements of the first column to the corresponding elements

3 −2 4

of the third column. In this case the determinant will look like this −2 1 1 = 2 ·

2 0 0

−2 4

= 2 · (−6) = −12.
1 1
Let’s formulate one more property.
10. The sum of multiples of elements of some row or column of the determi-
nant into their cofactors of the corresponding elements of other row or
column equals to zero.
Let’s prove one of possible equalities, for example, this one: a 11 ·A 21 +a 12 ·A 22 +a 13 ·A 23 =
0.
Let’s do evident transformations:

a 11 · A 21 + a 12 · A 22 + a 13 · A 23 = a 11 · (−M 21 ) + a 12 · M 22 + +a 13 · (−M 23 ) =

a 12 a 13 a 11 a 13 a 11 a 12
= −a 11 · + a 12 · − a 13 · = −a 11 · (a 12 · a 33 − a 13 · a 32 )+

a 32 a 33 a 31 a 33 a 31 a 32
+a 12 · (a 11 · a 33 − a 13 · a 31 ) − a 13 · (a 11 · a 32 − a 12 · a 31 ) = 0

Determinants of higher orders.
The determinants of not only second or third order, but of higher orders, are used in many

a 11 a 12 a 13 a 14

a 21 a 22 a 23 a 24
tasks. For example, the determinant of the forth order or the determinant

a 31 a 32 a 33 a 34

a 41 a 42 a 43 a 44

a 11 a 12 . . . a 1n

a 21 a 22 . . . a 2n
of n order: . It should be noted, that all properties shown above, could be
. . . . . . . . . . . .

a n1 a n2 . . . a nn
used for the determinants of higher orders as well. The rule of their calculation is limited to the
use of property 9 many times until the order of the determinants is lowered to the third, which
can be calculated by both property 9 and by the definition. Here is, it’s useful to use property 8
to get zero elements.
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