Page 14 - 4263

Matrices

A is a nonsingular matrix, so the inverse matrix exists.
Let’s calculate cofactors: A 11 = 13; A 12 = 11; A 13 = −3; A 21 = −9; A 22 = −1; A 23 = 12;
 
13 −9 16
A 31 = 16; A 32 = −3; A 33 = −7. So that, A −1 = 1  11 −1 −3  . The solution of the system
43
−3 12 −7
equals to:

     
13 −9 16 7 91 + 171 − 176
1 1
X = A −1 · B = ·  11 −1 −3   −19  = ·  77 + 19 + 33  =
·
43 43
−3 12 −7 −11 −21 − 228 + 77
   
86 2
1
= ·  129  =  3  .
43
−172 −4

2.5. Rank of a matrix

 
a 11 a 12 . . . a 1n
 a 21 a 22 . . . a 2n 
Let’s consider a rectangular matrix: A =   . If we choose any k rows
 . . . . . . . . . . . . 
a m1 a m2 . . . a mn
and k columns from this matrix, we will get a square matrix with k dimension and can calculate
the determinant of the last matrix. Such determinants are called minors of the given matrix.

Definition 2.2. The maximum order of minors not equal to zero is called the rank
of a matrix rank of a matrix. Thus, if the rank of a matrix equals to r, there is at least
one minor with r-order which doesn’t equal to zero, and at the same time all minors
of this matrix with (r + 1) order are equal to zero. The rank of a matrix is denoted
by r(A). ✓

In order to avoid computing a big number of determinants while calculating the rank of a
matrix, it is necessary to use basic (elementary) transformations, which are as follows:

1. Interchange two rows (or columns).
2. Multiply each element in a row (or column) by a non-zero number.
3. Multiply a row (or column) by a non-zero number and add the result to
another row (or column).
Matrices obtained from each other as a result of basic transformations are called equiva-
lent. Generally speaking, equivalent matrices are not equal to each other, but the ranks of those
matrices are equal .

Theorem 2.2.
The rank of a matrix doesn’t change when performing basic transformations. ⋆

The proof of this theorem is based on using corresponding properties of the determinants.

14