Can someone explain me to what to do here? I don't understand the question, or how to solve the problem. Should I use some theorems, to solve it?
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2$\begingroup$ Hint: On $AX=B$, to solve it algebraically would be to have $X=A^{-1}B$ while numerically is to actually compute what $X$ is (i.e., the numbers in $X$). $\endgroup$– Michael BurrMar 17, 2016 at 21:18
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$\begingroup$ Why A^-1 ? Do you have a link for additional information, to understand it better? $\endgroup$– AdiTMar 17, 2016 at 21:20
1 Answer
I agree with Micheal Burr.
One could also say, solve symbolically (by means of algebraic transformations) for the unknown $X$, e.g. $$ A^2 X + B = 0 \iff \\ X = (A^2)^{-1}(-B) = - (A^{-1})^2 B $$ and then insert the values for all variables and determine the resulting value for $X$, e.g. we enter $A$ and $B$ inta a computer algebra system (here Octave)
>> A = [ 2,2;3,4]
A =
2 2
3 4
>> B = [ 1,2;-1,2]
B =
1 2
-1 2
and then start calculating $X$:
>> inv(A)
ans =
2.0000 -1.0000
-1.5000 1.0000
>> X = - inv(A)*inv(A)*B
X =
-8.5000 -5.0000
7.0000 4.0000
A little test to see if the calculation worked:
>> A*A*X + B
ans =
1.4211e-014 0.0000e+000
2.8422e-014 1.4211e-014
and we are done.
The objects here are (square) matrices which form an algebraic structure called a ring. It means you can pretty much calculate with them like with numbers. Notable exceptions: multiplication is not commutative in general, not all elements have multiplicative inverses.
The task can be solved by knowing
- how to add and subtract matrices,
- how to multiply a matrix with a scalar,
- how to multiply matrices and
- how to calculate the multiplicative inverse of a matrix.
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$\begingroup$ Okay, I just ask again. Which rules of arithmetic do you use to isolate X? Where can I finde the information? Do you have a link? $\endgroup$– AdiTMar 17, 2016 at 21:32
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1$\begingroup$ We are using matrix algebra here. This is the subject of Linear Algebra. $\endgroup$– mvwMar 17, 2016 at 21:33