Invertible Equivalents  this sheet takes a given (invertible) matrix $A$ and describes several of the equivalent conditions on $A$.
Students should open this sage worksheet and work through the interactive cells below. Consider what Sage's conclusion in each case means. For small problems, try to work the problems also by hand to verify sage's conclusion. Answer (individually) the green questions posed in this worksheet. Note, that there are a few cells that pertain to stuff in chapter 3. Ignore these for now.
Finally, when you are finished with it all, go back through the sheet with a much larger value for n. Recompute all cells and see if the conclusions make sense conceptually. Discuss...
John Travis
Mississippi College
The user will first select the size for the $NxN$ matrix $A$ which will be used for the remainder of this page.
Enter N below. Click 'Update' twice to change the size.
If so, note that you will have to also reevaluate the other cells below.
Click to the left again to hide and once more to show the dynamic interactive window 
Now, enter the matrix $A$ below. You can also have the computer select one with rational entries at random. If you changed n above, you will likely need to reevaluate this cell first. This will require that you click on "hide" and then "ShiftEnter" or just quit the sheet and start over from the top.
Enter your data below. You can tab to navigate between cells. Click Update when you are finished.
If you want a random matrix, click the box and then update and then go on to the remainder of the page
Click to the left again to hide and once more to show the dynamic interactive window 
Here is the matrix you are using. Cutandpaste works well with this output. Here is the matrix you are using. Cutandpaste works well with this output. 
Save your Matrix to a File so that you can use the same one next time /home/sageserver/.sage/temp/travisPowerEdgeT430/19123MyMatrix.sobj Save your Matrix to a File so that you can use the same one next time /home/sageserver/.sage/temp/travisPowerEdgeT430/19123MyMatrix.sobj 
Reload your Matrix from the file... /home/sageserver/.sage/temp/travisPowerEdgeT430/19123MyMatrix Traceback (click to the left of this block for traceback) ... IOError: [Errno 2] No such file or directory: '/home/sageserver/.sage/temp/travisPowerEdgeT430/19123MyMatrix.sobj' Reload your Matrix from the file... /home/sageserver/.sage/temp/travisPowerEdgeT430/19123MyMatrix Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_19.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# * coding: utf8 *\\n" + _support_.preparse_worksheet_cell(base64.b64decode("cHJpbnQgIlJlbG9hZCB5b3VyIE1hdHJpeCBmcm9tIHRoZSBmaWxlLi4uIgpmaWxlbmFtZSA9ICdNeU1hdHJpeCcKZmlsZW5hbWVfZm9yX2xvYWRpbmcgPSBTQUdFX1RNUCtmaWxlbmFtZQpwcmludCBmaWxlbmFtZV9mb3JfbG9hZGluZwpBID0gbG9hZChmaWxlbmFtZV9mb3JfbG9hZGluZykKQQ=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in <module> File "/tmp/tmpCLaShh/___code___.py", line 6, in <module> A = load(filename_for_loading) File "sage/structure/sage_object.pyx", line 1015, in sage.structure.sage_object.load (/home/sageserver/sage7.5/src/build/cythonized/sage/structure/sage_object.c:12745) IOError: [Errno 2] No such file or directory: '/home/sageserver/.sage/temp/travisPowerEdgeT430/19123MyMatrix.sobj' 
Theorem 5
A matrix is nonsingular provided $Ax=b$ has exactly one solution for any $b$.
Click on Update to get a new vector b
Click to the left again to hide and once more to show the dynamic interactive window 

Assignment: Let sage give you vectors for b and the resulting solutions for x. Compute all your work on paper to turn in when I get back.
Theorem 7
A matrix is nonsingular provided it is row equivalent (and column equivalent) to the identity matrix.
Applying row reduction to A yieldsApplying row reduction to yieldsSince each of these is the identity, the equivalences are shown.
Applying row reduction to A yieldsApplying row reduction to yieldsSince each of these is the identity, the equivalences are shown.

Assignment:
A matrix is nonsingular provided the columns of $A$ span $R^n$. Thus, any vector b$\in R^n$ is in the span of the columns of $A$.
For the interactive cell below, note $A(2)$ refers to column 2 of the matrix $A$ for example.
Click on Update to get a new vector b
Click to the left again to hide and once more to show the dynamic interactive window 
Theorem 6
A matrix is nonsingular provided $A^{1}$ exists.
<p><font size=+2 color=blue><script type="math/tex">A^{1} =</script></font> <p><font size=+2 color=blue><script type="math/tex">A^{1} =</script></font> 
Theorem 7
A matrix is nonsingular provided $A^T$ is invertible.

Chapter 3
A matrix is nonsingular provided its determinant is nonzero.
The matrix A has determinant det(A) = 2
The matrix A has determinant det(A) = 2

A matrix is nonsingular provided $Ax=0$ has only the trivial solution $x=0$. That is, $Null(A)=\left\{{0 }\right\}$.

Save this for Chapter 3...
A matrix is nonsingular provided $A$ has no zero eigenvalues.
<font size=+2 color=blue>The eigenvalues for <script type="math/tex">A</script> =</font> <font size=+2 color=blue><b>They are all nonzero.</b></font> <font size=+2 color=blue>The eigenvalues for <script type="math/tex">A</script> =</font> <font size=+2 color=blue><b>They are all nonzero.</b></font> 
The matrix A can be expressed as a finite product of elementary matrices.
This one should be harder to determine (and render) and especially so for larger N.


