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Octave was made for linear algebra. The basics:

Matrix construction:

A = [1 2; 3 4]                  % 2x2
B = zeros(3)                     % 3x3 zeros
C = eye(4)                       % 4x4 identity
D = ones(2, 3)                   % 2x3 of ones
E = rand(3, 3)                   % 3x3 random in [0,1)

Element-wise vs matrix multiply:

A * B           % matrix multiplication
A .* B          % element-wise
A / B           % matrix divide (right): solves x*B = A
A \ B           % matrix divide (left): solves A*x = B  -- common for linear systems
A ./ B          % element-wise
A ^ 2           % matrix power
A .^ 2          % element-wise square

Transpose — A' (or A.' for non-conjugate). Important for complex.

Common operations:

det(A)             % determinant
inv(A)             % inverse
A\b                % solve A*x = b (better than inv(A)*b)
eig(A)             % eigenvalues
[V, D] = eig(A)    % eigenvectors V, eigenvalues D
rank(A)            % rank
norm(A)            % matrix norm
trace(A)           % sum of diagonal

Slicing:

A(1, :)            % first row
A(:, 2)            % second column
A(1:2, 2:3)        % submatrix
A(end, end)        % last element
A(end-1, :)        % second-to-last row

Reshape without copying data:

v = 1:12;
M = reshape(v, 3, 4)            % 3x4

Solving Ax = b is the killer feature — A\b uses LU decomposition under the hood, no manual algorithm needed.