https://preview.redd.it/9djx7dhp57s61.png?width=577&format=png&auto=webp&s=ecd6a1529af45285cb577fddc82eeaf51c413091

Tl;dr: Does there exist any applied applications to Cramer's rule? And are there enough good reasons for the students to learn it to keep it in the course compared to something else?

The standard linear algebra course at my university goes something like this:

• Gauss elimination
• Matrix Multiplication
• Inverting matrixes
• Solving linear systems
• Determinants / transpose
• Eigenvalues and eigenvectors
• Creating /changing Basis
• Doing the above using programming.

When lecturing inverting matrixes one of the themes in the curriculum is Cramer's rule. My problem with this is that I can not see for the life and death of me why I teach this to (the very applied focused) engineering students. It does terribly computationally. It is difficult to remember, compared to Gauss elimination. I have seen it used exactly once, and that was when showing that taking inverses in a matrix Lie group is smooth (which I do not see them doing anytime soon).

The only thing reason I can see for teaching it is that it gives a geometric interpretation of taking inverses. However, this takes time to explain, and for first-year students, most of it goes over their heads when trying.

Not a mathematician or math major but teaching myself some linear algebra. The link below has the part I'm a bit uncertain about and want to clear up before I get too far ahead.

https://imgur.com/a/w4cuwp0

My question is simply on why those other proceeding determinants go to 0 or "vanish". I believe it's because the aij in the 2nd to nth determinants represent redundant columns with respect to the cofactors. That would be the second thm referred to.

Is this correct? The first thm referred to is just that those product sums in parentheses are different representations of a determinant.

I'm in Linear Algebra 1, and having just covered Cramer's Rule, the prof showed this interesting case that I have a further question about the significance of.

Say we have a matrix containing a constant that can be adjusted, for instance, the system of equations
2cx+3y=6
4x+(c-1)y=4

giving the matrices

{{2c, 3}, {4, c-1}} {x, y} = {6, 4}

Since Cramer's rule only holds in cases where the determinant is nonzero, a typical question would be to find the values of c for which that is true. In this case, det(A)=0 when c=-2 or c=3. At c=-2 there are no solutions to the system, and at c=3 there are infinitely many solutions.

In the case of c=3, we cannot simply apply Cramer's rule, because the denominator of x=detA(1)/detA and y=detA(2)/detA are both detA=0.

However, what we can do is go back to the original system, leaving the variable c in the matrix, and calculate the values of detA, detA(1) and detA(2) in relation to c.

If I do that, and completely factor, I get:
detA=2(c+2)(c-3)
detA(1)=6(c-3)
detA(2)=8(c-3)

Now I can use limits to get an answer from the formulation of Cramer's rule in the case of c=3.

x= lim c→3 of 6(c−3)/(2(c+2)(c−3))

y= lim c→3 of 8(c−3)/(2(c+2)(c−3))

From which we can easily get the values of x=3/5 and y=4/5, which is a valid solution.

So Cramer's rule, despite its initial misgivings, has provided a solution to a system with a determinant of 0. My question (which my prof couldn't answer on the spot, which is why I'm bringing it here) is, which solution? What is special about these numbers, that they're the ones that happen to be spat out using this method? My first thought was that perhaps it's the particular solution, but it's not: the solution in parameterized form of the matrix when c=3 is

{x, y}={1, 0}+s{−1/2, 1}

So I am at a loss as to what these numbers "are," if they "are" anything in particular. Surely they're not just random?

(Also, if there's a better way to format matrices in markdown, please let me know!)

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-44t + d = 0

• 54 + d = -24.3

High schooler here,I encountered a physics problem which had 4 equations and 4 variables. Direct substitutions were too complicated so I figured maybe I could use Cramer's rule but I don't know how to solve a 4rth order determinant. I did ask my teacher about the physics problem and he did it using direct formulas not through assuming variables and my math teacher told me not to worry about higher orders till college.

In accordance with the title of this post, I would like to know which method is better and why? Also, under what conditions would one method be more efficient than the other. Thanks everyone.

At least where I'm from, even in engineering and other sciences, everyone learns Cramer's rule. I once had a teacher who said something along the lines of: "Of all the ways people have invented to solve linear systems, Cramer's rule is certainly the worst." and yet, as far as I know, it's taught everywhere. How come?

I have found this interesting method to finding the solutions of linear equations given n variables and solving 2^n-1 equations. This method reduces to Cramers Rule for 2 variables and 2 equations,

z=(x,y) and Az=B,

x=det[B , A*y* ]/detA

For n variables, Az=B

z=(x,y,...,z*n*)

And A is a 2^n-1 by n matrix,

I have found a method for finding z*n*

It involves constructing a ratio of nestled 2 by 2 matrix determinants. I'm not sure how to write matrices on here so this will be harder to describe.

For n=3, # of eqs =2^3-1 =4

z=(w, x, y)

x = det ({A*1* , B} {A 1 , A*3 * })/ det ({A*1* , A2 }{ A*1* , A*3* })

Where A*n* is the nth column

And { , } splits up the nestled two by two determinants inside of the larger determinant.

For n=5, 2^5-1 = 2^4 which means that for, say

x*5* = det B&A /det A

Where the top and bottom matrices are 16 by 16, which contain nestled 2 by 2 determinants of the constants which make up A (with an addition of B on the top), which are to be performed one from smallest to largest, in succession. So the the final solution for this variable, say, x is a are ratio of 2 by 2 determinants resembling Cramers rule.

Im not able to come up with the right typeset online. I'll follow up more later trying to perfectly describe this notation.

Has anyone seen this type of nestled determinants? I have tried looking everywhere. Maybe it isn't very useful, havent quite figure out the order of magnitude of the number of operations. I find the nestled determinants very unusual

A textbook I'm following told me to prove Cramer's rule (not an actual proof, just show that it works) with the following system of linear equations:

ax + by = e

cx + dy = f

I sent about solving the system through basic elimination and realized that I could potentially get two different answers for x and y (at least as far as I could tell). One was the answer you would get following Cramer's rule:

x = ed - bf / ad - bc AND y = af - ec / ad - bc

But, depending on which equation I multiplied by a negative variable (to eliminate one of them), I could also get:

x = bf - ed / bc - ad AND y = ec - af / bc - ad

I feel like I must be missing something as I should obviously not be getting the second set of equations. Any thoughts?

Could anyone explain proof derivation of Cramer Rule ?

https://math.stackexchange.com/a/1941610/511447 is confusing to me...

Someone told me the following, but I am more confused.

>det(x_1a_1 + x_2a_2 + x_3a_3 , a_2 , a_3)
>
>= det(x_1a_1 , a_2 , a_3) + det(x_2a_2 , a_2 , a_3) + det(x_3a_3 , a_2 , a_3)
>
>= x_1det(a_1 , a_2 , a_3) + x_2det(a_2, a_2, a_3) + x_3det(a_3, a_2, a_3)

Why det(a_2 a_2 a_3) must be 0 ?

Why det(a+b c d) = det(a c d) + det(b c d) ?

By the way, how does the 3-variables case work ?

https://preview.redd.it/kmo96pop07s61.png?width=577&format=png&auto=webp&s=172bedba8b4d42cefcacce3b283a491de51ede81

Using Cramer’s rule, solve the following system of equations:

𝑥 + 𝑦 + 𝑧 = 11, 2𝑥 − 6𝑦 − 𝑧 = 0, 3𝑥 + 4𝑦 + 2𝑧 = 0

https://www.desmos.com/calculator/mtzswssz7w

How can I put Cramer's rule into this desmos link? Can someone help me?

https://en.m.wikipedia.org/wiki/Cramer%27s_rule