How can I show these polynomials are not

Prove that in any group, an element and its inverse have the same order.

Let ${\displaystyle B=\{v1,v2,\dots ,vm\}}$ be a basis for $R^m$. Suppose kvm is a linear combination of $v1,v2,\cdots ,vm-1$ for some scalar k. What can be said about the possible value(s) of k?

struggling on the properties of the idempotent matrix, namely for any $n\times n$ matrix, $A^2=A.$. The projection matrix defined by $M=I_n-A{\left(A^{T}A\right)}^{-1}{A}^T$ is an idempotent matrix. The question is, for any given $n\times m$ ($n>m$) matrix $B,$, do we have
$\begin{array}{rcl}M& =& I_n-A{\left(A^{T}A\right)}^{-1}{A}^T\\ & =& I_n-AB{\left(B^T{A}^{T}AB\right)}^{-1}{B}^T{A}^T,\end{array}$
since $AB$ is basically the linear transformation of matrix $A.$

I don't claim at all to be an expert on this topic. In many (advanced) linear algebra textbooks for undergraduates, I usually find something about the "Jordan Canonical Form" of a matrix.
What is the purpose of such a form? I have taken a usual first-course in linear algebra (did another semester with Axler, but I don't claim to be an expert) and have taken abstract algebra (most familiar with group and ring theory) and have briefly skimmed through linear algebra books covering this material, but I don't quite understand the "big picture" idea, i.e., why is this useful in application? One person once told me it is the "most straightforward and useful algorithm for solving systems of linear equations, once you get beyond 3 variables or so," but maybe I'm missing something, since I usually don't see anything like what this person described to me in the linear algebra books I have. Most textbooks I've seen tend to have a more theoretical focus on this topic.
Also, any suggested texts which have good coverage on this topic would be very helpful.

Grades in three tests in College Algebra are 87,59 and 73.
The final carries double weight.
The average after final is 75
The points needed on the final to average the points to 75 when the final carries double weight.

Substitute each point (-3, 5) and (2, -1) into the slope-intercept form of a linear equation to write a system of equations. Then use the system to find the equation of the line containing the two points. Explain your reasoning.

Is the vector space $C\mathrm{\infty }[a,b]$ of infinitely differentiable functions on the interval [a,b], consider the derivate transformation D and the definite integral transformation I defined by $D(f)(x)=f\prime (x)D(f)(x)=f\prime (x)andI(f)(x)=\int xaf(t)dtf(t)dt$. (a) Compute $(DI)(f)=D(I(f))(DI)(f)=D(I(f))$. (b) Compute $(ID)(f)=I(D(f))(ID)(f)=I(D(f))$. (c) Do this transformations commute? That is to say, is it true that $(DI)(f)=(ID)(f)(DI)(f)=(ID)(f)$ for all vectors f in the space?