<s>[INST] finding determinant as an function in given matrix Calculate the determinant of the following matrix as an explicit
function of $x$. (It is a polynomial in $x$. You are asked to find
all the coefficients.)
\begin{bmatrix}1 & x & x^{2} & x^{3} & x^{4}\\
x^{5} & x^{6} & x^{7} & x^{8} & x^{9}\\
0 & 0 & 0 & x^{10} & x^{11}\\
0 & 0 & 0 & x^{12} & x^{13}\\
0 & 0 & 0 & x^{14} & x^{15}
\end{bmatrix}
Can someone help me with this question?
 [/INST] First, note that the 5th column is a multiple of the 4th column. That is,
\begin{bmatrix}
x^4\\
x^9\\
x^{11}\\
x^{13}\\
x^{15}\\
\end{bmatrix}
is $x$ times
\begin{bmatrix}
x^3\\
x^8\\
x^{10}\\
x^{12}\\
x^{14}\\
\end{bmatrix}.
Because the determinant of a matrix does not change when you subtract a multiple of one column from another column, we get that that matrix has the same determinant as that of
\begin{bmatrix}
1 & x & x^2 & x^3 & 0\\
x^5 & x^6 & x^7 & x^8 & 0\\
0 & 0 & 0 & x^{10} & 0\\
0 & 0 & 0 & x^{12} & 0\\
0 & 0 & 0 & x^{14} & 0\\
\end{bmatrix}
and you can easily tell that the determinant of that matrix is $0$.
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