Matrices in Matlab can be manipulated in many ways. For one, you can find the
transpose of a matrix using the apostrophe key:

C = B' C = 1 5 9 2 6 10 3 7 11 4 8 12

It should be noted that if C had been complex, the apostrophe would have
actually given the complex conjugate transpose. To get the transpose, use .'
(the two commands are the same if the matix is not complex).

Now you can
multiply the two matrices B and C together. Remember that order matters when
multiplying matrices.

D = B * C D = 30 70 110 70 174 278 110 278 446 D = C * B D = 107
122 137 152 122 140 158 176 137 158 179 200 152 176 200 224

Another option for matrix manipulation is that you can multiply the
corresponding elements of two matrices using the .* operator (the matrices must
be the same size to do this).

E = [1 2;3 4] F = [2 3;4 5] G = E .* F E = 1 2 3 4 F = 2 3 4 5 G = 2 6 12
20

If you have a square matrix, like E, you can also multiply it by itself as many
times as you like by raising it to a given power.

E^3 ans = 37 54 81 118

If wanted to cube each element in the matrix, just use the element-by-element
cubing.

E.^3 ans = 1 8 27 64

You can also find the inverse of a matrix:

X = inv(E) X = -2.0000 1.0000 1.5000 -0.5000

or its eigenvalues:

eig(E) ans = -0.3723 5.3723

There is even a function to find the coefficients of the characteristic
polynomial of a matrix. The "poly" function creates a vector that includes the
coefficients of the characteristic polynomial.

p = poly(E) p = 1.0000 -5.0000 -2.0000

Remember that the eigenvalues of a matrix are the same as the roots of its
characteristic polynomial: