The basic definitions
System from mn numbers (real, complex), either functions, or other objects, recorded in the form of the rectangular table consisting from m rows and n columns:
is called matrix .
Numbers (functions, other objects)
, making the matrix (1), are called
elements of a matrix
. Here the first index
i
designates the row number, and the second
j
– the column number on intersection of which the given element of a matrix is located.
or simply
. In this case speak, that the matrix
A
has dimension
m×n
. If
m=n
the matrix is called
square
of the order
n
. If
m≠n
the matrix is called
rectangular
. The matrix of dimension 1
×n
is called a vectorrow, and a matrix of dimension
m×
1 – a vectorcolumn. It is possible to consider usual number (scalar) as a matrix of dimension 1 × 1.
then it is called
diagonal
matrix.
Using Kronecker’s symbol
it is possible to record:
The concept determinant is connected with a square numerical matrix:
Matrix and its determinant different (though and connected) concepts. The numerical matrix A is the ordered system of numbers recorded in the form of the rectangular table, and its determinant det A is the number equal:
where the sum (4) extends on possible permutations of elements 1, 2..., n and, consequently, contains n ! addends, and k = 0, if the permutation is even and k = 1, if the permutation is odd.
S o l u t i o n . According to (3) we have:

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