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Operations with matrices

When all elements of a matrix, located below or above a main diagonal, are equal to zero,
then the matrix has a triangle view:
 
a 11 a 12 . . . a 1n
 0 a 22 . . . a 2n 
A =   .
 . . . . . . . . . . . . 
0 . . . 0 . . . a nn

When all elements of a matrix are equal to zero, then the matrix is called zero-matrix.
Two matrices A = (a ij ) and B = (b ij ) are equal (A = B), when all corresponding elements
are equal to one another, meaning a ij = b ij for any i and j. It is obvious, two matrices are equal
when they are of the same size.

2.2. Operations with matrices

1. Adding (subtracting) of matrices. A = (a ij ), B = (b ij ) — matrices.
A matrix, elements of which are equal to the sum of corresponding
elements, is called the sum of two matrices A and B of the same size. This
action is symbolised as: C = A + B.
The subs traction of two matrices is determined in the same way.
2. Multiplication of a matrix into a digit (number).
A matrix, each element of which equals to a multiple of the correspond-
ing element of the matrix into a digit is called the multiplication of the matrix
into a digit. This equation is symbolised like this: C = α · A.
For example, find matrix C, as a result of the following operations:
( ) ( ) ( ) ( ) ( )
2 0 1 2 6 0 −1 −2 5 −2
C = 3A−B = 3· +(−1)· = + = .
−1 1 3 1 −3 3 −3 −1 −6 2
3. Multiplication of matrices.
A multiple of a matrix into a matrix is called the matrix each element of which
equals to the sum of multiples of the elements of a row of matrix A
into the corresponding elements of a column of matrix B, such as: c ij =
a 11 ·b 1j +a 12 ·b 2j +. . .+a 1n ·b nj , (i = 1, 2, . . . , m; j = 1, 2, . . . , n). This is symbolised
as: C = A · B.
Here number k of columns of matrix A must be equal to the number of
rows of matrix B. Otherwise this operation is impossible.
It is necessary to say that if in adding matrices the inversion law is always
present, than in multiplication of matrices this law, generally speaking,
doesn’t work: A + B = B + A, but A · B ̸= B · A.
It is easy to check, that the following equations take place for the sum
and the multiplication of matrices:

(A + B) · C = A · C + B · C, A · (B · C) = (A · B) · C,

C · (A + B) = C · A + C · B, A + (B + C) = (A + B) + C.

Remark 2.1. If A and B are two square matrices of the same dimension,
and their determinants are equal to ∆A, ∆B, then the determinant of matrix
C = A · B equals to the multiple of determinants of these matrices: ∆C =
∆A · ∆B.

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