testMatrices.txt

The matrix multiplication problem below can be used to test the matrix browser.
select the meet of the matrix while in this file in an editor.
Now, press alt+shift+m. It should say initialised 2 by 3 matrix.
Select the next matrix, after the multiplication sign. 
Do the nvda+alt+m thing again.
You should hear selected 3 by 2 matrix.
Now, go below this matrix and start using the matrix browser to multiply the two matrices. You can directly type into the document as the browser doesn't interfeer with typing keys.
Keystrokes:
nvda+alt+m: initialised highlited matrix.
NVDA+alt+b: open matrix browser.
While in the browser, you can:

arrow through the current matrix.
press tab to view the next matrix.
press shift+tab to view the previous matrix.
press delete to delete a matrix.
Press escape to leave the browser.

$begin{Bmatrix}
1&2&3\\
4 & 6 & 8
\end{Bmatrix} \cdot \begin{Bmatrix}
8 & 5 \\
7 & 3 \\
1 & 8 
\end{Bmatrix}$

Matrix test 2.
Find the inverse of 
\begin{Bmatrix}
1 & 3 & 5 & 7\\
3 & 4 & 8 & 9 \\
8 & 5 & 8 & 3 \\
1 & 9 & 11 & 3
\end{Bmatrix}
matrix test 3: from a homework assignment.
\textbf{Exercise 0.0.6:}

In class we saw that if $E_{i\longleftrightarrow j}$ denotes the matrix
which has entry $1$ at position $(k,k)$ for $k\neq i$
, $j$; entry $1$ at positions $(j,i)$ and $(i, j)$ and $0$ everywhere else (So it is the identity matrix,
with rows $i$ and $j$ swapped), then multiplication on the left by $%
E_{i\longleftrightarrow j}$ corresponds to the row operation of
interchanging rows $i$ with $j$:

$E_{i\longleftrightarrow j}\left( 
\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1n} \\ 
\vdots & \vdots & \ddots & \vdots \\ 
a_{i1} & a_{i2} & \cdots & a_{in} \\ 
\vdots & \vdots & \ddots & \vdots \\ 
a_{j1} & a_{j2} & \cdots & a_{jn} \\ 
\vdots & \vdots & \ddots & \vdots \\ 
a_{n1} & a_{n2} & \cdots & a_{nn}%
\end{array}%
\right) =\left( 
\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1n} \\ 
\vdots & \vdots & \ddots & \vdots \\ 
a_{j1} & a_{j2} & \cdots & a_{jn} \\ 
\vdots & \vdots & \ddots & \vdots \\ 
a_{i1} & a_{i2} & \cdots & a_{in} \\ 
\vdots & \vdots & \ddots & \vdots \\ 
a_{n1} & a_{n2} & \cdots & a_{nn}%
\end{array}%
\right) $

\vspace{1pt}

Let $E_{ia}$ be the matrix that has $1$s on the diagonal, except at position 
$\left(i, i\right) $ where it has an $a$; and $0$ everywhere else:

$E_{ia}=\left( 
\begin{array}{cccc}
1 & 0 & \cdots & 0 \\ 
0 & 1 & \cdots & 0 \\ 
\vdots & \vdots & \ddots & \vdots \\ 
0 & \cdots & a & \cdots \\ 
\vdots & \vdots & \ddots & \vdots \\ 
0 & 0 & \cdots & 1%
\end{array}%
\right) $

Prove that multiplying an $n\times m$ matrix $A$ on the left by $E_{ia}$ has
the effect of multiplying the $i$-th of $A$ by $a$. So $E_{ia}$ corresponds
to the row operation of scaling.

\vspace{1pt}

Finally, let $E_{i+j}$ be the matrix that has $1$s at the diagonal, one more 
$1$ at position $(j, i)$, and all other entries $0$.
Prove that multiplyng $A$ on the left by $E_{i+j}$ corresponds to adding the 
$i$-th row of $A$ to the $j$-th row of $A$.

\vspace{1pt}

The upshot of the exercise is that Gaussian elimination can be expressed as
multipling a matrix by an appropriate sequence of the elementary matrices $%
E_{i\longleftrightarrow j}$, $E_{ia}$, $E_{i+j}$.