## Properties of Determinants Part 6

more about this property…

P9: The determinant of a product is the product of the determinants.

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# Tag: matrices

## Properties of Determinants Part 6

## Properties of Determinants Part 5

## Properties of Determinants Part 4

## Properties of Determinants Part 3

## Properties of Determinants – Part 2

## Elementary Matrices in Mathematica

more about this property…

P9: The determinant of a product is the product of the determinants.

really practical property

P9: The determinant of a product is the product of the determinants.

yet another property of determinants

P8: If a matrix is singular its determinant is zero, if it is invertible then its determinant is not zero.

Important Consequence: The determinant is plus or minus the product of its pivots.

two more properties of determinants

P6: If a matrix contains a row of zeros then its determinant is zero.

P7: If a matrix is triangular, then its determinant is the product of the diagonal entries.

Two more properties:

P4: If two rows are equal then the determinant is zero.

P5: Adding a multiple of one row to another row doesn’t change the determinant.

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

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 [math]2\times 2[/math]. 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 that multiply the matrix to produce a lower triangular matrix.

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

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

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
```

One final matrix

```
E32 = {{1,0,0},{0,1,0},{0,-2,1}};
E32 . E31 . E21 . A//MatrixForm
```

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

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

for any ! All we have to do is setup the augmented matrix 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

Let’s now take a particular vector

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.