## Elementary Matrices in Mathematica

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You can enter a matrix in Mathematica in a number of ways. One way is to type up the entries as a nested list with each sublist serving as the rows of the matrix. For example the matrix

$Latex formula$

can be entered as

A={{1,2,3},{4,5,6},{7,8,0}}

You can also enter a matrix using the Basic Math Assistant Palette (Paletts -> Basic Math Assistant).

By default, the matrix that appears is $2\times 2$. If you want to add more rows then you can use keyboard shortcuts CTRL + ENTER and to add more columns use CTRL + , (comma). Here is a link to a wolfram tutorial.

Whether you define a matrix as a nested list, using keyboard shortcuts, or any other method, MatrixForm changes the output format to look like standard mathematical typsetting. For example once the matrix A is define with the code above, the code

A//MatrixForm

or

MatrixForm[A]

will produce a nicely typeset version.

Now we are going to talk about using elementary matrices $Latex formula$ that multiply the matrix $Latex formula$ to produce a lower triangular matrix.

The matrix $Latex formula$ will subtract 4 times the first row from the second row:

$Latex formula$

enter this in Mathematica as

E21={{1,0,0},{-4,1,0},{0,0,1}};

E21//MatrixForm

This code has two lines, the first line defines the matrix E21 but suppresses output because the line ends with a semicolon. The second line display this matrix with nice typesetting. We will now multiply these two matrices:

E21 . A

produces

{{1, 2, 3}, {0, -3, -6}, {7, 8, 0}}

You may or may not put a space before and after the dot, I used a space to make the code more readable. Once again, using MatrixForm we can get a nicer version of the output:

MatrixForm[E21 . A]

At this point we have

$Latex formula$

Now define the matrix E31 that subtracts 7 times the first row from the third row:

E31 = {{1,0,0},{0,1,0},{-7,0,1}};

E31 . E21 . A//MatrixForm

$Latex formula$

One final matrix

E32 = {{1,0,0},{0,1,0},{0,-2,1}};

E32 . E31 . E21 . A//MatrixForm

$Latex formula$

Let’s now define a new matrix F that takes care of all of these row operations at once.

F = E32 . E31 . E21;

F//MatrixForm

F . A//MatrixForm

These three lines of code will produce

$Latex formula$

$Latex formula$

This is good stuff! This new matrix F immediately does all row operations and gives us an upper triangular matrix. The great thing is, we can now solve the matrix equation

$Latex formula$

for any $Latex formula$! All we have to do is setup the augmented matrix $Latex formula$ and multiply this matrix on left with F:

Ab = {{1,2,3,b1},{4,5,6,b2},{7,8,0,b3}};

Ab//MatrixForm

F . Ab //MatrixForm

These three lines of code give

$Latex formula$

Let’s now take a particular vector

$Latex formula$

and apply F to the augmented matrix:

{b1,b2,b3} = {4, 3, 5};

MatrixForm[Ab]

MatrixForm[F . Ab]

This code produces an updated version of the augmented matrix and the result of applying F to it.

Here is a link to a Mathematica Notebook showing the computations of this post. Here is a link to the PDF version of the same file.