Linear algebra

System of equations
Coefficient matrix
Right-hand side values
Augmented matrix
Swap rows 1 and 3
Eliminate values in rows 2 and 3
Divide through second row
Row operation
Divide through third row
Add rows
Add scaled rows
Matlab can computer determinants automatically
Matlab can calculate the row reduced echelon form
Matrix division

System of equations

In linear algebra a system of equations can be represented by a coefficient matrix. MATLAB can solve all types of linear equations automatically. We will demonstrate how MATLAB matrix manipulation commands can be used to perform row operations on a system of linear equations.

Consider the system:

Coefficient matrix

We define a standard coefficient matrix.

A = [4 3 1; 5 4 2; 1 -1 1]

A =

     4     3     1
     5     4     2
     1    -1     1

Right-hand side values

These are the values from the right hand side of the equations.

C = [10;4;15]

Augmented matrix

Row reduction is done on the augmented matrix.

AA = [A C]

AA =

     4     3     1    10
     5     4     2     4
     1    -1     1    15

Swap rows 1 and 3

The syntax manipulates two rows of the matrix at the same time by indexing them in a matrix. The colon indicates all columns.

AA([1 3], :) = AA([3 1], :)

AA =

     1    -1     1    15
     5     4     2     4
     4     3     1    10

Eliminate values in rows 2 and 3

We use a trick to manipulate two rows at the same time. The final term in the command is a 2x1 matrix multiplied by a 1x4 matrix. This produces two copies of the first row, each scaled differently.

AA([2 3],:) = AA([2 3],:) - [5;4]*AA(1,:)

AA =

     1    -1     1    15
     0     9    -3   -71
     0     7    -3   -50

Divide through second row

AA(2,:) = AA(2,:) ./ 9

AA =

    1.0000   -1.0000    1.0000   15.0000
         0    1.0000   -0.3333   -7.8889
         0    7.0000   -3.0000  -50.0000

Row operation

AA(3,:) = AA(3,:) - 7 * AA(2,:)

AA =

    1.0000   -1.0000    1.0000   15.0000
         0    1.0000   -0.3333   -7.8889
         0         0   -0.6667    5.2222

Divide through third row

AA(3,:) = AA(3,:) ./ -0.6667

AA =

    1.0000   -1.0000    1.0000   15.0000
         0    1.0000   -0.3333   -7.8889
         0         0    1.0000   -7.8329

Add rows

AA(1,:) = AA(1,:) + AA(2,:)

AA =

    1.0000         0    0.6667    7.1111
         0    1.0000   -0.3333   -7.8889
         0         0    1.0000   -7.8329

Add scaled rows

AA([1 2], :) = AA([1 2], :) + [-0.6667; 0.3333] * AA(3,:)

AA =

    1.0000         0   -0.0000   12.3333
         0    1.0000   -0.0000  -10.4996
         0         0    1.0000   -7.8329

MATLAB can computer determinants automatically

The "det" command will compute a determinant.

det(A)

ans =

     6

MATLAB can calculate the row reduced echelon form

A much simpler approach is to just ask MATLAB to compute the row reduced echelon form of the augmented matrix.

rref(AA)

ans =

    1.0000         0         0   12.3333
         0    1.0000         0  -10.5000
         0         0    1.0000   -7.8333

Matrix division

MATLAB supports a form of matrix division. In the equation:

Solving for X gives:

MATLAB can perform this calculation with "left division":

A\C

ans =

   12.3333
  -10.5000
   -7.8333

Published with MATLAB^® 7.4

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