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Matrices

The second case. ∆ = 0. Suppose at least one of the auxiliary determinants ∆ 1 , ∆ 2 , ∆ 3 doesn’t equal to
zero. In that case at least one of the equalities (1.7) will be impossible, meaning that system (1.7) and (1.6)
have no solution. In case if ∆ = ∆ 1 = ∆ 2 = ∆ 3 = 0 then the system either doesn’t have any solution or
has lot’s of them. It’s recommended to study this case individually.

Lecture 2. Matrices

2.1. Definition of a matrix

A set of numbers a ij , located in a rectangular table, that consists of mrows and n columns, is
called a matrix. Numbers a ij are called as elements or entries of a matrix. A matrix is denoted
 
a 11 a 12 . . . a 1n
 a 21 a 22 . . . a 2n 
by a capital letter A, B, C, etc: A =   . The size of the above matrix
 . . . . . . . . . . . . 
a m1 a m2 . . . a mn
is m × n. Like in a determinant, each element of a matrix has its own place, marked by double
indexation (i — the row number, j — the column number).
Supposing m = 1, we have a matrix-row: A = (a 11 a 12 . . . a 1n ); n = 1, we have a matrix -
   
a 11 a 11 a 12 . . . a 1n
 a 21   a 21 a 22 . . . a 2n 
column: A =   . When m = n, we have a square matrix: A =   .
 . . .   . . . . . . . . . . . . 
a m1 a n1 a n2 . . . a nn
A square matrix can be matched with a determinant, which is written:

a 11 a 12 . . . a 1n

a 21 a 22 . . . a 2n
det A = .
. . . . . . . . . . . .

a n1 a n2 . . . a nn
When det A ̸= 0, the matrix is called a nonsingular matrix, and a singular one on the contrary.
The main diagonal of a matrix consists of elements with equal indexes a 11 , a 22 , . . . , a nn .
These elements are called as diagonal in any matrix.
A square matrix, which consists of diagonal elements not equal to zero, and other elements
equal to zero, is called a diagonal matrix:

 
0 . . . 0
a 11
 0 a 22 . . . 0 
A =   .
 . . . . . . . . . . . . 
0 . . . . . . a nn

When all diagonal elements of such a matrix are equal to unity, then a diagonal matrix transforms
into a unit matrix, which is denoted by

 
1 0 . . . 0
 0 1 . . . 0 
E =   .
. . . . . . . . . . . .
 
0 0 . . . 1

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