# Questions tagged [determinant]

Questions about determinants: their computation or their theory. If $E$ is a vector space of dimension $d$, then we can compute the determinant of a $d$-uple $(v_1,\ldots,v_d)$ with respect to a basis.

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### Can Cramer's Rule really distinguish between infinite no. of solutions and no solution?

This is a question which was asked in a high-school exam held in India(JEE ADVANCED). Going by Cramer's rule, for infinite solutions, I should get $D=D_1=D_2=D_3=0$ (where $D$ is the original ...
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### Proof of determinants for matrices of any order

I was told that the determinant of a square matrix can be expanded along any row or column and given a proof by expanding in all possible ways, but only for square matrices of order 2 and 3. Is a ...
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### Let $X$ and $Y$ be two matrices of order $3\times 2$ and $2\times 3$. Then Least value of $|YX|$ is

Let $X$ and $Y$ be two matrices of order $3\times 2$ and $2\times 3$ respectively such that $XY=\begin{bmatrix} 2 & -2 & 0\\ -2 & 2 & 0\\ 0 & 0 &2\\ \end{bmatrix}$ Then ...
### Determine the entries $x$ and $y$ in a matrix so that its only eigenvalue is $1$.
I am doing some self-study in preparation for an exam, and in this problem I am given the following matrix in $R^{3×3}$: $\begin{pmatrix} 1&0&1\\ 0&1&-1\\ 0&x&y\\ \end{pmatrix}$...
### $A,B \in GL_{n}$. Let $f(x) = det(xA+(1-x)B)$. Then conclude $f$ is a non constant polynomial.
The Actual Question $A,B \in GL_{n}$. Let $f(x) = det(xA+(1-x)B)$. Then conclude $f=0$ has finitely many solutions. Thoughts I understand that $f$ is a non zero polynomial, and if it's not a constant ...