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Matrices

Let’s consider the example of multiplying two matrices:

   
1 2 −1 5 −3 2
C = A · B =  3 0 2   4 0 1  =
·
−2 1 1 1 2 2
 
1 · 5 + 2 · 4 + (−1) · 1 1 · (−3) + 2 · 0 + (−1) · 2 1 · 2 + 2 · 1 + (−1) · 2
=  3 · 5 + 0 · 4 + 2 · 1 3 · (−3) + 0 · 0 + 2 · 2 3 · 2 + 0 · 1 + 2 · 2  .
−2 · 5 + 1 · 4 + 1 · 2 −2 · (−3) + 1 · 0 + 1 · 2 −2 · 2 + 1 · 1 + 1 · 2

2.3. Inverse matrix

Definition 2.1. For a given square matrix A, matrix A −1 is called an inverse ma-
trix, if it verifies the following conditions: A · A −1 = A −1 · A = E, where E - is a unit
matrix. ✓

Theorem 2.1.

In order for square matrix A to have the inverse one, it is necessary and enough for matrix A
to be nonsingular, i.e. its determinant doesn’t equal to zero. ⋆

PROOF. Necessity. Let’s assume that there is an inverse matrix for matrix A. Let’s show that in
this case matrix A has to be nonsingular, meaning ∆A ̸= 0. Indeed, if ∆A = 0, then the deter-
−1
minant of the multiple: ∆(A · A ) = ∆A · ∆A −1 = 0, which is impossible, as according to the
−1
definition: A · A −1 = E and ∆(A · A ) = ∆E = 1. We will prove sufficiency on the example of
a matrix of the third order:
 
a 11 a 12 a 13
A =  a 21 a 22 a 23  .
a 31 a 32 a 33
Let’s prove, that if ∆A ̸= 0, then the inverse for matrix A is such a matrix:

 
A 11 A 21 A 31
1
A −1 = ·  A 12 A 22 A 32  (2.1)
∆A
A 13 A 23 A 33
which consists of cofactors of the elements of matrix A, written in a transposition order.
Following the definition, let’s check, that A · A −1 = E. Indeed, using properties 9 and 10 of the
determinants, we have:
     
1 a 11 a 12 a 13 A 11 A 21 A 31 1 ∆A 0 0
A · A −1 = ·  a 21 a 22 a 23   A 12 A 22 A 32  = ·  0 ∆A 0  =
·
∆A ∆A
a 31 a 32 a 33 A 13 A 23 A 33 0 0 ∆A
 
1 0 0
=  0 1 0  = E.
0 0 1

In an analogy, it is possible to prove, that A −1 · A = E. So, the inverse matrix A −1 for matrix A
of the third order is determined by formula (2.1). 2

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