# Tagged Questions

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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### For any $A, B \in SL(2, F)$, does knowing $\operatorname{tr}A$, $\operatorname{tr}B$, and $\operatorname{tr}AB$ specify $A$ and $B$?

In title, $F$ denotes a field. Does knowing the trace of two matrices and their product specify those two matrices? Up to some equivalence, perhaps?
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### Question about row operations and row-echelon form,

If I have a matrix, with, say, the first two columns consisting of all zeroes, then is the first entry of the third column, which is non-zero, my first pivot variable, so that when solving Ax=b, for x=...
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### Matrix Properties Problem

If $A\in M(n\times n;R)$ and $K= \dfrac {A+A^T}{2}$ and $L= \dfrac{A-A^T}{2}$. Prove: i) that $K$ and $-L$ are symmetric ii) that $K+L=A$ iii) that $K$ and $L$ are unique matrices with the properties ...
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### If $\det(A)=0$, must the null space of $A$ be zero? [closed]

Came along this question: If $\det(A)=0$ for an $N\times N$-dimensional matrix $A$, the null space of $A$ is equal to zero. True or false? Why? Thank you already!
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### Why the determinant of a matrix with the sum of each row's elements equal 0 is 0?

I'm trying to understand the proof of a problem, but I'm stuck. In my book they consider that if all lines of a matrix has sum 0 then it's determinant is also 0. I checked some random examples and it'...
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### How to solve the matrix equation $A\overrightarrow{x}=\overrightarrow{b}$ in Matlab when nullitity$(A)\neq 0$

Say, $A= \begin{pmatrix} 1 & 0 &1 \\ 0 & 1 &1 \\ 0 &0 &0 \end{pmatrix}$ and $\overrightarrow{b}= \begin{pmatrix} 8 \\ -5 \\ 0 \end{pmatrix}$ and I want to solve ...
### How to solve linear system of form $(A \otimes B + C^{T}C)x = b$ when $A \otimes B$ is too large to compute?
For the given linear system: $$(A \otimes B + C^{T}C)x = b$$ where $\otimes$ is the Kronecker product, $A$ and $B$ are dense and symmetric positive-definite, and $C^{T}C$ is a sparse symmetric block ...
Let $P_{\pi}$ denote a permutation matrix associated to the permutation $\pi:\{1,...,n\}\rightarrow \{1,...,n\}$ and $\sigma$ denote the cyclic permutation $(1 2 ...n)$. If T is the $n\times n$ lower ...