# Tagged Questions

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### Determine the values of $k$ so that the following linear system has unique, infinite and no solutions.

Determine the values of $k$ so that the following linear system has a unique solution, infinite solutions and no solution. $2x + (k + 1)y + 2z = 3$ $2x + 3y + kz = 3$ $3x + 3y − 3z = 3$ I have ...
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### Find matrix determinant

How do I reduce this matrix to row echelon form and hence find the determinant, or is there a way that I am unaware of that finds the determinant of this matrix without having to reduce it row echelon ...
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### Find Determinant of A

I've tried creating a triangular matrix, tried row reducing but can't figure it out as I keep on having c-unknown in my answer. How would I do this?
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### Given the matrix, find c such that det(D)=0 has a repeated solution

Given the matrix $D= \begin{vmatrix} 1 & x & x^2\\ 2 & c & 4\\ 3 & 2 & 1 \end{vmatrix}$ Find c such that $\det(D)=0$ has a repeated solution for $x\in R$. I got up to ...
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### Find signature of symmetric block matrix, given the diagonal blocks are positive / negative definite - Check my proof

This may be a basic question, but I'd like someone to double check it. We are given the matrix $A=\begin{pmatrix} A_1 & C \\ C^T & A_2\end{pmatrix}$ where $A_1$ is a $k$ by $k$ positive ...
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### Difficulty proving formula containing the adjugate and determinant of a matrix

This is what I need to prove: You have an invertible matrix $A \in M_3(\Bbb R^3)$. Prove that $\operatorname{adj}(\operatorname{adj}(A))=\det{(A)}^{n-2}A$ The proof goes as follows: ...
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### $T=-T^{*}$, show that $T+\alpha I$ is invertible.

Please don't answer the question. Just tell me if I am in the right direction. I should be able to solve this. We are given $T=-T^{*}$, show that $T+\alpha I$ is invertibe for all real alphas that ...
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### $n\times n$ matrix determinant of rook configuration
We place $n$ rooks on an $n\times n$ chessboard in such a way that they don't threaten each other. To each such placement corresponds an $n\times n$ matrix in which there is a $1$ at the position of ...