Matrix associative property

[cooking71]

I will give the example from the text as I do not understand how they come up with what they did. As I cannot type the brackets each set of numbers will be it's own matrix.

If
A = 1 -2
-3 4


B = 3 0 -1
1 1 2


C = 1 0
0 2
1 1


compute ABC in 2 ways


Grouping BC gives A(BC) =

1 -2 ( 3 0 -1 1 0 )
-3 4 1 1 2 0 2
1 1



= 1 -2 2 -1 = -4 -9
-3 4 3 4 6 19


I hope you can understand what I typed as in the box will not keep the matrices as I have them typed I did the best I could. My problem is that the text goes right to the answer yet does not show what to multipy by what. if you can tell me that I can undertand the second part of the example as well. As an online student I just do not have access to a professor and finding this site I hope will be a huge help.

THank you in advance.

K

 

[soroban]

Hello, cooking71!

\(\displaystyle A\:=\:\begin{pmatrix}1 & -2 \\ -3 & 4\end{pmatrix}\;\;\;B \:= \:\begin{pmatrix}3 & 0 & -1 \\ 1 & 1 & 2\end{pmatrix}\;\;\;C\;=\;\begin{pmatrix}1 & 0 \\ 0 & 2 \\ 1 & 1\end{pmatrix}\)

Compute \(\displaystyle ABC\) in two ways.

\(\displaystyle (AB)C\;=\;\left[\begin{pmatrix}1 & -2 \\ -3 & 4\end{pmatrix}\cdot\begin{pmatrix}3 & 0 & -1\\ 1 & 1 & 2\end{pmatrix}\right]\,\cdot\,\begin{pmatrix}1 & 0 \\ 0 & 2 \\ 1 & 1\end{pmatrix} \;= \;\begin{pmatrix}1 & -2 & -5 \\ -5 & 4 & 11\end{pmatrix}\,\cdot\,\begin{pmatrix}1 & 0 \\ 0 & 2 \\ 1 & 1\end{pmatrix} \;= \;\begin{pmatrix}-4 & -9 \\ 6 & 19\end{pmatrix}\)

\(\displaystyle A(BC) \;= \;\begin{pmatrix}1 & -2 \\ -3 & 4\end{pmatrix}\,\cdot\,\left[\begin{pmatrix}3 & 0 & -1 \\ 1 & 1 & 2\end{pmatrix}\cdot\begin{pmatrix}1 & 0 \\ 0 & 2 \\ 1 & 1\end{pmatrix}\right] \;=\;\begin{pmatrix}1 & -2 \\ -3 & 4\end{pmatrix}\,\cdot\,\begin{pmatrix}2 & -1 \\ 3 & 4\end{pmatrix} \;=\;\begin{pmatrix}-4 & -9 \\ 6 & 19\end{pmatrix}\)

 

[cooking71]

That is exactly what the text has but I do not understand how it got to

2 -1
3 4


When I try to do the mulitpication myself I am not coming up with that answer. I need to understand what is being multipied and how.

Thank you.


K

 

[soroban]

Hello, cooking71!

Do you mean you don't know how to multiply matrices?
\(\displaystyle \;\;\)You've never heard of "row times column"?

I hope I'm wrong . . .


\(\displaystyle BC \;= \;\begin{pmatrix}3 & 0 & -1 \\ 1 & 1 & 2\end{pmatrix}\,\cdot\,\begin{pmatrix}1 & 0 \\ 0 & 2 \\ 1 & 1\end{pmatrix} \;=\;

\begin{pmatrix}(3)(1)+(0)(0)+(-1)(1) & (3)(0)+(0)(2)+(-1)(1)\\ \qquad\qquad(1)(1)+(1)(1)+(2)(1)\;\; & (1)(0)+(1)(2)+(2)(1)\end{pmatrix}\)

\(\displaystyle \;\;= \;\begin{pmatrix}3\,+\,0\,-\,1 &\qquad 0\,+\,0\,-\,1 \\ 1\,+\,0\,+\,2 &\qquad 0\,+\,2\,+\,2\end{pmatrix} \;= \;\begin{pmatrix}2 & -1 \\ 3 & 4\end{pmatrix}\)