Matrices and systems of equations

[Avatrin]

Hi

I am now filling in what I perceive to be gaps in my knowledge. One of these problems is understanding why matrices can solve systems of equations. I do completely get Gaussian elimination to solve systems of linear equations. However, when using determinants and the like to solve, for instance, systems of differential equations, I do not feel I have properly learned why that works.

Whenever I pick up a book on systems theory, I am reminded that I understand metric spaces and Lebesgue integration better than something as basic as a matrix. I need to change that.

What book can, through a rigorous manner, explain matrix theory to me?

 

[Fredrik]

Almost any book on linear algebra will do. "Linear algebra done wrong" by Sergei Treil recently became my favorite linear algebra book.

 

[HallsofIvy]

Perhaps it would help to look at a very simple example. Suppose we have the equations ax+ by= c and dx+ ey= f. I decide to eliminate "y" and solve for x. Multiply the first equation by "e": aex+ bey= ce. Multiply the second equation by b: bdx+ bey= bf. Now that y has the same coefficient in each equation, we eliminate y by subtracting: (aex+ bey)- (bdx+ bey)= (ae- bd)x= ce- bf. So, as long as ae- bd is not 0, x= (ce- bf)/(ae- bd).

You can see that this is the same as "As long as $\left|\begin{array}{cc}a & b \\ d & e\end{array}\right|$ is not 0,
$$x= \frac{\left|\begin{array}{cc} c & b \\ f & e \end{array}\right|}{\left|\begin{array}{cc}a & b \\ d & e\end{array}\right|}$$".