What exactly are matrices and determinants?

[A.MHF]

I'm taking a Differential Equations class and we're dealing with matrices and determinants. I've dealt with them before but I was always annoyed by the fact that I don't know what the heck is going on. So I know that matrices are a way to organise linear equations and make transformations easier, but I don't understand why they're the way they are. Why for example are the input data written vertically and the operations horizontally? Wouldn't it be easier if both of them had the same alignment? Did a bunch of mathematicians just decide that this is the way they're going to be?
And also, what's with the determinants? How on Earth did we get the formula ad-bc for the 2x2 square matrices? What do they represent, and why is it messy to calculate the determinant for 3x3? And finally, how is it possible to solve system of equations, find volumes, and cross multiply vectors using them? I mean, these things are barely related, yet they can all be solved similarly.
Please help me understand all of this, it'd would also be great if someone could provide a material to the history of all of this.

Thanks.

 

[Mark44]

I'm taking a Differential Equations class and we're dealing with matrices and determinants. I've dealt with them before but I was always annoyed by the fact that I don't know what the heck is going on. So I know that matrices are a way to organise linear equations and make transformations easier, but I don't understand why they're the way they are. Why for example are the input data written vertically and the operations horizontally?

A matrix is a rectangular array of numbers, symbols, or expressions (cribbed from Wikipedia - https://en.wikipedia.org/wiki/Matrix). There is nothing in the definition of a matrix that specifies anything about input data or operations, so I don't know what you're referring to here.

A system of linear equations can be represented in a shorter form as an augmented matrix. The same types of operations that you would perform on pairs of equations can be performed on the rows of a matrix, yielding a matrix that is "equivalent to" the starting matrix. IOW, the solution that is represented by the final matrix is also the solution to the starting matrix. Using matrices in this manner is mostly a matter of convenience -- you don't have to write so much stuff.

Wouldn't it be easier if both of them had the same alignment? Did a bunch of mathematicians just decide that this is the way they're going to be?
And also, what's with the determinants? How on Earth did we get the formula ad-bc for the 2x2 square matrices?

Someone along the way defined it to be like this.

What do they represent, and why is it messy to calculate the determinant for 3x3?

The determinant of a square matrix is a measure of a matrix, in some sense. More formally, the determinant is a mapping from the set of n x n matrices to the real numbers. If you think the formula for the determinant of a 3 x 3 matrix is messy, the formulas for 4 x 4, 5 x 5, and higher order matrices are even worse. It is possible, though, to use a few properties of the determinant to simplify these calculations. Also, you can expand a determinant by minors, breaking down, say, the determinant of a 4 x 4 matrix into four 3 x 3 determinants.

One of the most useful properties of the determinant is that if its value is zero, the associated matrix is noninvertible.

And finally, how is it possible to solve system of equations, find volumes, and cross multiply vectors using them?

Cramer's Rule uses determinants to solve a system of equations. The cross product, which is limited exclusively to 3-D vectors, uses a pseudo-determinant to arrive at the vector that represents the cross product of the two input vectors. It's not really a determinant, which can be inferred from the "pseudo" in the description (a true determinant evaluates to a number, not a vector), but the calculation is done in the same way. The magnitude of the cross product, $|\vec{A} \times \vec{B}|$, gives the area of the parallelogram determined by the two vectors. There's another product involving vectors that gives the volume of the box that the three vectors determine.

I mean, these things are barely related, yet they can all be solved similarly.
Please help me understand all of this, it'd would also be great if someone could provide a material to the history of all of this.

Wikipedia is a good starting point. Search for "matrix", "determinant", and "Cramer's Rule".

 

[Geofleur]

Regarding the determinant question: Take three vectors and write their components down as column matrices. Now stick them into a matrix. The determinant of the matrix is the volume of the parallelipiped that has those three vectors along its sides. It's a signed volume because, depending on the order of the vectors, you may end up with negative the volume instead. This geometric interpretation of determinants can be very useful, because you can use it to calculate volumes in higher dimensional spaces. If you think of an area as a "2 dimensional volume", you can use determinants to get areas too!

Regarding the alignment question: Matrices are often used to represent more abstract entities called linear operators. Operators act on things, and usually we think of operators acting on things placed toward the right of them. So we think of matrices as operators that act on things placed on their right hand sides. Consequently, a vector will need to be represented as a column with m rows so that an n x m matrix can "act" on it. We can, of course, think of operators as acting on things placed on their left sides instead. If we had chosen to do that, then vectors would be expressed as rows instead of columns.

 

[WWGD]

In the most general sense, a matrix is just an array of numbers. In specific, given situations, it will take a given meaning. It can serve to, e.g., represent a system of linear equations, a (finite-dimensional *) linear operator, as the adjacency graph of a matrix, as representing a Markov process, etc.

* Though there are infinite matrices too.

It seems, from the context of your question, that your matrices are used to represent linear operators. Like others said, determinants are used to decide whether a set of linear equations are linearly dependent from each other. For the $2 \times 2$ -case, a line is given by a pair of numbers (these numbers determine the slope of the line). Then the two lines are independent if the slopes are equal. This is all equivalent to having $ad-bc=0$. Something similar for $n \times n$ matrices.

 

[mathwonk]

an m by n matrix represents a linear map from R^n to R^m.. Hence a row vector of length n is a linear map from R^n to R^1, or a real valued linear function on R^n, while a column vector of length n is a linear map from R^1 to R^n. Since that is determined by where it sends the number 1, this is equivalent to a vector in R^n. That is why a vector is written vertically and a real valued linear function is written horizontally, so you can distinguish the functions from the vectors they act on.

another aspect of the determinant is that it is the denominator that will show upo if you solve an n by n system of linear equations, hence it must bne zero if the solution is to be unique. try it by solving a 2 by 2 system and see if you get ad-bc in the bottom of your fractional answers.

 

[WWGD]

an m by n matrix represents a linear map from R^n to R^m.. Hence a row vector of length n is a linear map from R^n to R^1, or a real valued linear function on R^n, while a column vector of length n is a linear map from R^1 to R^n. Since that is determined by where it sends the number 1, this is equivalent to a vector in R^n. That is why a vector is written vertically and a real valued linear function is written horizontally, so you can distinguish the functions from the vectors they act on.

another aspect of the determinant is that it is the denominator that will show upo if you solve an n by n system of linear equations, hence it must bne zero if the solution is to be unique. try it by solving a 2 by 2 system and see if you get ad-bc in the bottom of your fractional answers.

Isnt it more accurate to say that every matrix is equivalent to a linear map ; more formally, I think this is equivalent to the fact that the space of all linear maps (we can even define it between modules, but it is much clearer if we work with $\mathbb R^n, \mathbb R^m$ ) defined on an ordered basis is isomorphic to the set of all $n \times m$ matrices under multiplication (with the j-th column being the image of the $j_th$ vector in an ordered basis )? A matrix may represent many other things depending on the context. If we have , e.g., a graph, the matrix may define the connectedness properties of the graph, by using a $0-1$ -matrix $M_{ij}$, where $m_{ij}=1$ if there is an edge joining vertex $i$ with vertex $j$ and $m_{ij}=0$ otherwise.