Ed Bueler's Matrices in Matlab: Basic

Matlab is designed to make manipulating matrices and solving linear systems easy. For instance, the next few lines enter and solve the linear system A x = b: 
» A=[2 -3 2; -4 2 -6; 2 2 4]

A =

     2    -3     2
    -4     2    -6
     2     2     4

» b=[5 14 8]'

b =

     5
    14
     8

» A\b

ans =

  1.0e+002 *

   1.09000000000000
   0.27000000000000
  -0.66000000000000
The second input shows a standard trick for entering column vectors without too many keystrokes: enter the vector as a row vector, and the take its transpose.

Note that the answer is x1 = 109, x2 = 27, and x3 = -66, but Matlab frequently writes computed quantities in scientific notation.

Calculating inverses and determinants is straightforward: 

» inv(A)

ans =

   5.00000000000000   4.00000000000000   3.50000000000000
   1.00000000000000   1.00000000000000   1.00000000000000
  -3.00000000000000  -2.50000000000000  -2.00000000000000

» det(A)

ans =

     4
Extracting a row and substituting a column (for instance) are easy operations if one takes advantage of Matlab's colon notation: 
» A(2,:)

ans =

    -4     2    -6

» AA=[A(:,1:2) b]

AA =

     2    -3     5
    -4     2    14
     2     2     8
Matrix multiplication, transpose, and "applying a matrix to a vector" (which is also matrix multiplication) look like: 
» A*AA

ans =

    20    -8   -16
   -28     4   -40
     4     6    70

» A'

ans =

     2    -4     2
    -3     2     2
     2    -6     4

» bb=b'

bb =

     5    14     8

» A*b

ans =

   -16
   -40
    70
But matrix multiplication must be done with matrices of the appropriate dimensions, and Matlab enforces it: 
» A*bb
??? Error using ==> *
Inner matrix dimensions must agree.
Useful commands include quick entry of larger identity and zero matrices, and augmenting one matrix by another: 
» eye(4,4)

ans =

     1     0     0     0
     0     1     0     0
     0     0     1     0
     0     0     0     1

» zeros(2,3)

ans =

     0     0     0
     0     0     0

» M=[A eye(3,3)]

M =

     2    -3     2     1     0     0
    -4     2    -6     0     1     0
     2     2     4     0     0     1