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9h
comment Row sum of $P^{m}$ when row sum of $P$ is $1$
To me the row sum of a $n\times n$ matrix is $n$-tuple of numbers, not one number. Even then it would be good to be clear whether you mean the "sum of the rows" (a row obtained by summing each column) or the "vector of rows sums" (which is the sum of the columns).
16h
comment Matrix diagonalization theorems and counterexamples: reference-request.
The fundamental question here is to broad for this forum. As for the reference request, very few linear algebra books provide tables of all combinations of question one might ask with their answers (the theory just does not work that way).
16h
revised Assigning a specific value to components of a vector
edited tags
16h
comment Assigning a specific value to components of a vector
You might explain that $i,j,k$ are vectors of a standard basis (unit vectors). Otherwise your question makes little sense, and it even looks like you are talking about quaternions.
17h
revised Calculate the product of these $98$ rational numbers
Somewhat more informative title
2d
comment proof-similar matrices have the same characteristic polynomial
@gbox: Yes that is true. Although I might pedantically add to the statement about scalar matrices "as long as scalars themselves commute". If ever you consider matrices over non-commutative rings (which is perfectly possible) then "scalar matrices" no longer always commute, because left-multiplication by a given scalar may be different from right-multiplication by it. But then scalar matrices are not a very useful notion is such a setting to begin with.
2d
comment proof-similar matrices have the same characteristic polynomial
This looks needlessly complicated to me. Why not just $(xM)I=xM=I(xM)$? Also your second display never uses commutation of $xM$ with $I$, indeed it never even forms $xM$ to begin with. Instead it uses the (unproved here) fact that $(xA)B=A(xB)$, with $A=M^{-1}$ and $B=I$.
2d
answered proof-similar matrices have the same characteristic polynomial
2d
comment Why should quaternions exist?
In point 1. you are giving a wrong reasoning that moreover is in the wrong context. For the latter, reasoning about numbers of equations and unknowns arises in the context of linear equations, and these are not linear equations. But anyway having more equations than unknowns does not imply much if the equations are dependent, and these equations could well be dependent (I think maybe $k^2=-1$ could be deduced from the other equations, though I did not check).
Aug
26
comment Binomial Sum Formula
@TerraHyde: Although I'd have no difficulty with this problem, I would find it very hard to formulate a partial solution. Once you've got the idea, it is bang on.
Aug
26
comment Binomial Sum Formula
@Lucian: more precisely, this can be rewritten to one side of the Vandermonde's identity; OP is asking about the expression at the other side of it (which of course is easy to obtain from your link).
Aug
26
revised Can someone check my answers on group permutation and answer part (g)
added 65 characters in body
Aug
26
comment Is 4 the second or third digit of pi
I know all ten digits that occur in the decimal expansion of $\pi$. I'm pretty sure there aren't any others.
Aug
26
comment Is 4 the second or third digit of pi
One cannot actually measure any circle around the observable universe. Besides general relativity gets very important as such scales, and it gets hard if not impossible to even define what a circle is, what its circumference and diameter are, and even then the formula from Euclidean plane geometry is no longer valid.
Aug
25
comment Find a matrix of the linear map in the given basis
The concrete values of the $Y$ basis vectors indeed do not influence the answer, due to the way the problem is formulated. In fact you don't even need to know the the space at departure is $\Bbb R^3$, just that $Y=[y_1,y_2,y_3]$ is some basis of it.
Aug
25
comment Find a matrix of the linear map in the given basis
The quoted text is fairly clear. The linear transformation is given by its images of the vectors $y_i$ (not of the standard bases), and you need to find its matrix w.r.t. the bases $Y$ and $W$. Since the initial information already give the matrix $\pmatrix{4&1&7\\5&3&1}$ of $T$ w.r.t. the bases $Y$ (not the standard basis!) and the standard basis of $\Bbb R^2$, you only need to perform change of basis at arrival.
Aug
25
answered a matrix of rank $r$ satisfies a polynomial of degree $r+1$.
Aug
25
revised a matrix of rank $r$ satisfies a polynomial of degree $r+1$.
added 1 character in body
Aug
25
comment Find a matrix of the linear map in the given basis
It is unclear what you are doing. Notably the values for $w_1$ and $w_2$ do not appear to be used at all.
Aug
25
comment probability that 5 square lie along a diagonal line - doubt
This is particularly obvious in the context of "square chosen from a chessboard", where clearly the board is viewed as a collection of squares to choose from.