We perform operations on matrices and compare different ways of dealing with linear systems of equations.

The goals of this tutorial is to add, to multiply and to invert matrices.

NOTE: This tutorial is available in Swedish: MATLAB_TUTO2_Sv.pdf

Matrix algebra

Just as for vectors, the addition of matrices and the multiplication of matrices with scalars are defined by element. In Matlab, these operations are performed with +, - and * as usual. Unlike vectors, we can also multiply matrices if their dimensions match. If $LaTeX: A=\left(a_{ij}\right)\:\in\mathbb{R}^{n\times m}$ and $LaTeX: B=\left(b_{ij}\right)\:\in\mathbb{R}^{m\times r}$ , $LaTeX: AB \in\mathbb{R}^{n\times r}$ and its elements are given by: $LaTeX: (AB)_{i,j}=\sum_{k=1}^{m} a_{ik} b_{kj}$ .

This is the "row-column rule" and is executed with * .

Matlab also has elemental multiplication $LaTeX: (AB)_{i,j}=a_{ij} b_{ij}$ . This is performed with .* : A.*B .

This is useful in programming, but we do not study this product in this course.

Practice exercise 1:

Define your own matrices A and B of dimension 2 x 3 , C of dimension 3 x 3 , D of dimension 1 x 3 and F and G of dimension 3 x 1 .

a) Check that the dimensions are correct with the command size .

b) Try to calculate A + B, −A, 2A, 3A - 5B and A + C. Explain any error messages.

c) Try to calculate AC , AG , and AB . Explain any error messages.

d) Compare DG and GD .

e) Calculate C²and C¹⁰.

More matrix operations

The transpose A^T of a matrix A is written in Matlab as A ' . Example:

Screenshot 2022-01-25 at 10.25.05.png

This is especially useful for ensuring that all variables that are intended to represent vectors are interpreted as either column or row vectors. If you start mixing, problems arise quickly. You can also quickly define a column vector by e.g. x = [1 2 3 4] ' and thus avoid all semicolons.

The inverse A ^{− 1} of a matrix A is calculated using the command inv (preferred) or simply A ^ -1 . Example:

Screenshot 2022-01-25 at 10.25.15.png

If the matrix does not have an inverse, you get a warning, but gives also an answer (which does not mean anything):

Screenshot 2022-01-25 at 10.25.21.png

Practice exercise 2:

Let A and G be the matrices from Practice exercise 1.

a) Calculate A^T and compare with A.

b) Add a third row in A by setting it equal to G^T. Let A be the new matrix.

c) Calculate A ^{− 1}. (If it does not have an inverse, change some of the elements in A.)

d) Test via the definition that what was calculated is really the inverse of A.

e) Verify the law of arithmetic $LaTeX: (A^{T})^{-1}= (A^{-1})^{T}$ for this specific matrix A.

Systems of equations

You can use the second matrix inverse to solve certain linear systems of equations. If $LaTeX: A\in \mathbb{R} ^{sn\times n}$ (square matrix) and LaTeX: Ax=b have a unique solution, this is given by $LaTeX: x=A^{-1} b$ .

If there are several solutions or no solutions at all, A^-1 is not defined.

This can be useful if you have to solve many systems with the same matrix A but with different right-hand links b.

Otherwise, however, it is always better to use the row reduction method (Gauss elimination) instead. This requires fewer calculations and introduces minor errors (from roundings, right-hand errors, etc.). In the case of many right-hand links, we can factorize the matrix A as LU, instead of forming the inverse (we will see this method in details later in the course).

Solving LaTeX: Ax=b in Matlab via row reduction can be done in several steps via rref as we saw last week, but also directly via x = A \ b. If A is not square, you get a so-called minimum square solution instead, which we will study later in the course. If the system is not solvable, such a "solution" is the best thing that can be achieved. It is worth mentioning that the syntax A / B, matrix division from the right, which corresponds to AB^{− 1}, i.e. multiplication with the inverse from the second direction, is also mentioned.

Practice exercise 3:

Repeat the second practice exercise from Week 2 - Matlab tutorial, but use \ and inv instead of rref. Compare the results with what you got from rref.