Matrices and systems of equations

1. Feb 27, 2015

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?

2. Feb 27, 2015

Fredrik

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

3. Feb 28, 2015

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,
$$x= \frac{\left|\begin{array}{cc} c & b \\ f & e \end{array}\right|}{\left|\begin{array}{cc}a & b \\ d & e\end{array}\right|}$$".