LINEAR ALGEBRA (Math 211)
Sheet #2

Multiplication of matrices. \fbox{${\cal M}(m,n)\times {\cal M}(n,k) \to {\cal M}(m,k)$}


\begin{displaymath}\begin{picture}
(500,140)(0,0)
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...95,87){\line(0,1){50}}
\put(390,140){\fbox{$t$}}
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Matrix representation of a linear system:         \fbox{$A\cdot X=B$}

\begin{displaymath}\left(\begin{array}{rrrr}
a_{11}&a_{12}&\dots&a_{1n}\\
a_{21...
...ft(\begin{array}{c} b_1\\ b_2 \\ \vdots\\ b_m\end{array}\right)\end{displaymath}

Indentity matrix: $\displaystyle I=\left(\begin{array}{cccc}
1&0&\dots&0\\
0&1&\dots&0\\
\vdots&\vdots& &\vdots\\
0&0&\dots&1
\end{array}\right) \in {\cal M}(n,n)$.
Inverse matrix: \fbox{$A\cdot A^{-1} = A^{-1}\cdot A =I$}

$\displaystyle \left(\begin{array}{cc}
a&b\\
c&d
\end{array}\right)^{-1} = {1\over ad-bc} \left(\begin{array}{rr}
d&-b\\
-c&a
\end{array}\right)$


Solving a sytem $A\cdot X=B$ of n linear equations of n variables:         \fbox{$X= A^{-1}\cdot B$}





An elementary matrix is a matrix which can be obtained from an identity matrix by one of the following tree operations:

$\textstyle \parbox{10cm}{
{\bf (1)} Multiply a row by a nonzero constant.\\
{\...
...} Interchange two rows.\\
{\bf (3)} Add a multiple of one row to another row.}$

Theorem. If E is an elementary matrix then the matrix $E\cdot A$ is obtained from matrix A by the row operation (1)-(3) corrsponding to E.



Examples.

$\left(\begin{array}{rr}
-5&0\\
0&1
\end{array}\right)$
  $\left(\begin{array}{rrrr}
0&0&1&0\\
0&1&0&0\\
1&0&0&0\\
0&0&0&1
\end{array}\right)$   $\left(\begin{array}{rrr}
1&0&0\\
0&1&0\\
7&0&1
\end{array}\right)$
$\uparrow$  $\uparrow$  $\uparrow$
Multiplication of the first row by -5  Interchanging of the first and third rows  Addition 7 times the first row to the third row


A method for inverting matrices.

$\bullet$ Adjoin the identity matrix I to the right side of A producing $\left( A \vert I \right)$.

$\bullet$ Apply row operations (1)-(3) to this matrix until the left half is reduced to I. The right half will be converted into the inverse matrix: $\left( I \vert A^{-1} \right)$.

LINEAR ALGEBRA (Math 211)
HW #3 (Due Mon. Sept. 25 )


Multiply the following matrices.

1. $\left(\begin{array}{rr}
1&2\\
3&4
\end{array}\right)\cdot\left(\begin{array}{rr}
5&6\\
7&8
\end{array}\right) =\ ?$ 2. $\left(\begin{array}{rrrr}
9&-3&5&6\\
6&-2&3&4\\
3&-1&3&14
\end{array}\right)\cdot\left(\begin{array}{r}
2\\
-7\\
-7\\
0
\end{array}\right) =\ ?$


Find the following inverses.

3. $\left(\begin{array}{rr}
\cos\alpha & \sin\alpha\\
-\sin\alpha& \cos\alpha
\end{array}\right)^{-1} =\ ?$


4. $\left(\begin{array}{rrr}
1&0&1\\
1&1&0\\
1&1&1
\end{array}\right)^{-1} =\ ?$
5. $\left(\begin{array}{rrrr}
a_{11}& 0&\dots&0\\
0&a_{22}&\dots&0\\
\vdots&\vdots&\ddots&\vdots\\
0& 0&\dots&a_{nn}
\end{array}\right)^{-1} =\ ?$, if $a_{11}\cdot a_{22}\cdot\dots\cdot a_{nn}\not= 0$.










LINEAR ALGEBRA (Math 211)
HW #4 (Due Wed. Sept. 27)


Find the following inverses and their representation as the product of elementary matrices.

1. $\left(\begin{array}{rr}
-3& 8\\
-1& 3
\end{array}\right)^{-1}$ 2. $\left(\begin{array}{rr}
7& 9\\
-6&-8
\end{array}\right)^{-1}$




Find the inverses.

3. $\left(\begin{array}{rrr}
2&7&3\\
3&9&4\\
1&5&3
\end{array}\right)^{-1}$ 4. $\left(\begin{array}{rrrrr}
1&1&1&\dots&1\\
0&1&1&\dots&1\\
0&0&1&\dots&1\\
\vdots&\vdots&\vdots&\ddots&\vdots\\
0&0&0&\dots&1
\end{array}\right)^{-1}$


5. Solve matrix equation         $\left(\begin{array}{rr}
3&-1\\
5&-2
\end{array}\right)\cdot X\cdot
\left(\begi...
...\end{array}\right) =
\left(\begin{array}{rr}
14&16\\
9&10
\end{array}\right)$