Linear Systems and Matrices

Linear Systems

An n by n linear system of equations is a system of n linear equations in n variables.

a11x1 + a12x2 + ... + a1nxn  =  b1
a21x1 + a22x2 + ... + a2nxn  =  b2
...            ...                ...        ...
an1x1 + an2x2 + ... + annxn  =  bn

Example

Solve

2x1 + 3x2  =  9
x1 - 2x2  =  1

Solution

To solve this we sequentially perform members of the following three operations:

1. Switch two equations.

2. Multiply an equation by a nonzero constant.

3. Replace an equation by that equation plus a multiple of the second equation.

We have

Switching the two equations
x1 - 2x2  =  1
2x1 + 3x2  =  9

Replace the 2nd equation with the 2nd equation + (-2)1st equation
x1 - 2x2  =  1
7x2  =  7

Multiply the second equation by 1/7
x1 - 2x2  =  1
x2  =  1

Replace the 1st equation with the 1st equation + (2)2nd equation

x1   =  3
x2  =  1

Matrices

An m by n matrix is an array of numbers with m rows and n columns.

Example

The matrix below is a 2 by 3 matrix. A square matrix is an n by n matrix, that is a matrix such that the number of rows is equal to the number of columns.  The ijth entry is the number in the ith row and jth column.  For example, the the matrix above the 1 2th entry is

a12  =  4

Note:  A vector such at <2,4,6> can be looked as a 1 by 3 matrix.

A square matrix is called a diagonal matrix if

aij  =  0        for        i j

The matrix below is a diagonal matrix If all the entries of a diagonal matrix are equal, then the matrix is called a scalar matrix.  The example below is a scalar matrix. Just as with vectors we can add and subtract matrices and multiply a matrix by a scalar.  To add or subtract matrices the dimensions of the two matrices must be the same.

 Definition Let A and B be m by n matrices and k be a scalar then         (A + B)ij  =  Aij + Bij        (A - B)ij  =  Aij - Bij       (kA)ij  =  kAij

Example

Let then Two matrices are called equal if all of their entries are equal.

If A is an m by n matrix, then the transpose of A, AT, is the n by m matrix with the rows and columns switched.

(AT)ij  =  Aji

In the above example Example of the Theoretical Exercise

Prove that

(AT)T  =  A

Solution

We have

((AT)T )ij =  (AT)ji  =  Aij

Since the ijth entries are equal for each ij, the matrices are equal.