Thread: Matrices & Systems of Equations:Matrix Algebra

1. Matrices & Systems of Equations:Matrix Algebra

Hello could someone please help me with the following problems, I just couldn't find my way around it. Thanks!

1) Given the following:
A equals
1 2
1 -2

b equals
4
0

c equals
-3
-2

write c as a linear combination of column vectors a_1 & a_2

2) Let A equals

a_11 a_12
a_21 a_22

Show that if d = a_11 a_22 - a_21 a_12 is not equal 0, then
A inverse equals

1/d * ( {a_22 -a_12} {-a_21 a_11} ) ====> {row 1} {row 2}

2. Originally Posted by googoogaga
Hello could someone please help me with the following problems, I just couldn't find my way around it. Thanks!

1) Given the following:
A equals
1 2
1 -2

b equals
4
0

c equals
-3
-2

write c as a linear combination of column vectors a_1 & a_2
Suppose $x~\bold{a_1}+y ~\bold{a_2}=\bold{c}$, then we may write this as a system of linear equations:

$
x+2y=-3
$

$
x-2y=-2
$

which you can now solve for $x$ and $y$.

RonL

3. Originally Posted by googoogaga
2) Let A equals

a_11 a_12
a_21 a_22

Show that if d = a_11 a_22 - a_21 a_12 is not equal 0, then
A inverse equals

1/d * ( {a_22 -a_12} {-a_21 a_11} ) ====> {row 1} {row 2}
This requires that you multiply the matrices:

$
\left( \begin{array}{cc} a_{11}&a_{12}\\a_{21}&a_{22} \end{array} \right)
$

and:

$
\frac{1}{d}\left( \begin{array}{cc} a_{22}&-a_{12}\\-a_{21}&a_{11} \end{array} \right)
$

to show that this is the 2x2 identity matrix

RonL

4. Matrices and Systems Equations

I still don't get number two. Once multiplying the matrices, then where do I go from there? And what is the 2*2 identity matrice?

5. Originally Posted by googoogaga
I still don't get number two. Once multiplying the matrices, then where do I go from there? And what is the 2*2 identity matrice?
You need to show that
$\frac{1}{a_{11}a_{22} - a_{12}a_{21}}\left ( \begin{matrix} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{matrix} \right ) \cdot \left ( \begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} \right ) = \left ( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right ) = I_2$

-Dan