# Algebraic Formulas

## Matrix. Determinants

**Definition. **A **matrix** is a rectangular table of numbers.

* ** *

E.g. . We can define a matrix thus:

Let *M*={*1, 2, 3, ..., m*} and *N=*{*1, 2, 3, ..., n*}. *A*: *M *x *N -> ***R, ***A(i,j) = a*_{i,j} is called matrix type (*m, n*), with *m *rows and *n *columns.

A matrix that has a size equal to one is called *vector*. A matrix A[1,n] (1 row, *n *columns) is called** line vector **and matrix B[m,1] ( one column, *m *rows) is called **column** **vector**.

Examples:

It is a matrix type 4x3. The item *A[3,1]* or *a*_{3,1} is *12*.

is a matrix type (1, 7) or **row matrix**.

A matrix A(*m,n*) which has *m = n * is called *square matrix*. So a **square matrix** is the matrix that has the number of lines equal to the number of columns.

**Matrix addition**

If* A *and *B *are two matrix type *m *x *n, *then *C = A *+ *B, *where *c*_{i,j} = a_{i,j} +* b*_{i,j} is the **sum** (where *i<m+1, j<n+1*).

Exemple:

**Multiplication by a constant**

Having the matrix *A * and the constant *c, *we have *B = cA, *where *b*_{i,j} = ca_{i,j} which is the multiplication of matrix *A* by the constant *c*. For instance,

**Multiplication**

Let the matrix *A *type *m *x *n* and *B* a matrix type *n *x *p*. Then their product is *C = AB* a matrix type *m* x *p*, with

*c*_{i,j} = a_{i,1}b_{1,j} + a_{i,2}b_{2,j} + ... + a_{i,n}b_{1,n}. For instance,

**The properties of matrix multiplication**

1. - associativity

2. - neutral element, where* I*_{n} is the unit matrix is defined as

3.

4. - distributivity.

A square matrix *A* of order n is **inversely **(or non-singular) if there is a square matrix* B* of order *n*, so, to have

AB = I_{n} = BA

In this case, the matrix **B** is called reverse of the matrix **A**, and is denoted **A**^{-1}.

**Determinants**

Let be the matrix

. It is called the *determinant* of the matrix A, the number

We write .

**Keywords: **
algebra, formulas, matrix, matrices, determinants