1  Matrices and Systems of Linear Equations

When solving systems of linear equations the essential information of the system can be recorded compactly in a rectangular array called a matrix. We show it on an example.

Example Given the system of linear equations

3 x 1 + x 2 4 x 3 + x 4 = 1 x 1 + 2 x 3 x 4 = 0 2 x 1 3 x 2 + 3 x 4 = 1 (1.1)

coefficients of each variables are aligned in columns and we get

( 3 1 4 1 1 0 2 1 2 3 0 3 ) (1.2)

so called the matrix of the system (1.1).

If we add the right hand side as the last new column we get

( 3 1 4 1 1 0 2 1 2 3 0 3 | 1 0 1 ) (1.3)

so called augmented matrix of the system (1.1).

Let us give a brief introduction what a matrix is.

A matrix A is an arrangement of m.n numbers as follows:

A = ( a 1 , 1 a 1 , j a 1 , n a i , 1 a i , j a i , n a m , 1 a m , j a m , n ) . (1.4)

A matrix of the form (1.4) is said to be of the type m×n, of an m×nmatrix. It has m rows and n columns; the entry ai,jin the i-th row and j-th columns is called the (i,j)-entry of A. For instance, the (2,4)-entry is the number a2,4.

The columns of the matrix A are

( a 1 , 1 a i , 1 a m , 1 ) , ( a 1 , 2 a i , 2 a m , 2 ) , , ( a 1 , n a i , n a m , n ) ;

the rows of A are

( a 1 , 1 , , a 1 , j , , a 1 , n ) , , ( a m , 1 , , a m , j , , a m , n ) .

For a short we often write A=(ai,j) instead of (1.4) making sure that the type of the matrix is clear from the context.

Example Consider the following matrix

A = ( 1 2 1 3 2 3 0 1 2 2 1 0 0 3 4 ) . (1.5)

A is a matrix of type 3×5. Its columns are

( 1 3 1 ) , ( 2 0 0 ) , ( 1 1 0 ) , ( 3 2 3 ) , ( 2 2 4 ) ;

its rows are

( 1 , 2 , 1 , 3 , 2 ) , ( 3 , 0 , 1 , 2 , 2 ) , ( 1 , 0 , 0 , 3 , 4 ) .

An (i,i)-entry ai,iis called a diagonal entry; the diagonal of the matrix A is a1,1,a2,2,,an,nfor nm, and a1,1,a2,2,,am,m for nm.

The diagonal entries of the matrix (1.5) are: a1,1=1, a2,2=0, and a3,3=0.

If a matrix has the same number of rows and columns, i.e. if it is n×n matrix, then we call it a square matrix.

A matrix A=(ai,j) is called upper triangular if all its entries bellow its diagonal are 0. More precisely, an m×nmatrix A=(ai,j) is upper triangular if ai,j=0 for all i>j.

A matrix A=(ai,j) is called lower triangular if all entries above its diagonal are 0. More precisely, an m×nmatrix A=(ai,j) is lower triangular if ai,j=0 for all i<j.

A square matrix which is both upper triangular and lower triangular is called a diagonal matrix. In other words, a matrix A=(ai,j) is diagonal if ai,j=0 for all ij.

The matrix (1.5) is neither upper triangular nor lower diagonal. An example of an upper triangular matrix is the matrix B, an example of a lower triangular matrix is the matrix C where

B = ( 1 0 3 7 0 0 2 3 0 0 1 2 ) , C = ( 1 0 0 3 0 0 1 2 1 ) .