Linear Algebra

Solving a Linear System

A linear algebraic equation is an equation of the system

a₁ x₁+a₂ x₂+a₃ x₃+⋯+a_n x_n=b

where a's are constant coefficients, the x's are the unknowns, and b is a constant. A solution is a sequence of numbers s₁,s₂, and s₃that satisfies the equation.

Example

4x₁+5x₂-2x₃=16

is such an equation in which there are three unknown: x₁,x₂,andx₃. One solution to this equation is x₁=3,x₂=4 and x₃=8,since 4*3+5*4-2*8 is equal to 16.

A system of a linear algebraic equation is a set of the equation of the form:

a₁₁ x₁+a₁₂ x₂+a₁₃ x₃+⋯+a_1n x_n=b₁
a₂₁ x₁+a₂₂ x₂+a₂₃ x₃+⋯+a_2n x_n=b₂
a₃₁ x₁+a₃₂ x₂+a₃₃ x₃+⋯+a_3n x_n=b₃
a_m1 x₁+a_m2 x₂+a_m3 x₃+⋯+a_mn x_n=b_m

This is called an m*n system of equations; there are m equations and n unknowns.

Matrix Forms

Because of the method that matrix multiplication works, these equations can be defined in the matrix form as Ax = b where A is the matrix of the coefficients, x is the column vector of the unknown, and b is a column vector of the constant from the right-hand side of the equations:

A           x      =       b
a₁₁ a₁₂ a₁₃...a_1n       x₁       b₁
a₂₁ a₂₂ a₂₃...a_2n       x₂       b₂
a₃₁ a₃₂ a₃₃...a_3n       x₃       b₃
.............................       .............       ........
a_m1 a_m2 a_m3...a_mn       x_n       b_m

A solution set is a set of all possible solutions to the system of equation (all sets of value for the unknowns that solve the equation). All systems of linear equations have either:

• No solutions
• One solution
• Infinitely many solutions

Solution using matrix Inverse

Probably the simple way of solving this system of equations is to use the matrix inverse.

A^-1 A=1

We can multiply both sides of the matrix equation AX= B by A^-1to get

A^-1 AX=A^-1 B

Or

X=A^-1 B

So, the solution can be found as a product of the inverse of A and the column vector b.

In MATLAB, there are two method of doing this, using the built-in inv function and matrix multiplication, and also using the "\" operator:

>> A = [3 4 1; -2 0 3; 1 2 4]	
A =
        3     4      1
      -2      0      3
       1      2      4	
>> b = [2 1 0]'	
     b =
          2
         1
         0  
 >> x = inv(A) * b
x =
       -1.1818
        1.5000
       -0.4545
>> A\b
ans =
      -1.1818
       1.5000
       -0.4545

Solving 2x2 Systems of Equations

The simplest system is a 2 x 2 system, with just two equations and two unknowns. For these systems, there is a simple definition for the inverse of a matrix, which uses the determinant D of the matrix.

For a coefficient matrix, A generally defined as

the determinant D is defined as a₁₁ a₂₂-a₁₂ a₂₁

Example

          x₁+3x₂=-2
          2x₁+4x₂=1

This would be written in matrix form as

The determinant D = 1*4 -3*2 = -2.

MATLAB has built-in functions det to find the determinant of the matrix.

>> A = [1 3; 2 4]
A =
         1      3
         2      4
>> b = [-2;1]
b =
        -2
         1
>> det(A)
ans =
          -2
>> inv(A)
ans =
         -2.0000     1.5000
          1.0000     -0.5000
>> x = inv(A) * b
x =
      5.5000
    -2.5000