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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5
votes
9answers
275 views
+100

Shortest and most elementary proof that the product of an $n$-column and an $n$-row has determinant $0$

Let $\bf u$ be any column vector and $\bf v$ be any row vector, each with $n \geq 2$ arbitrary entries from a field. Then it is well known that ${\bf u} {\bf v}$ is an $n \times n$ matrix such ...
0
votes
0answers
32 views
+50

Given $A\in Tn(R)$ show thast A is a scalar matrix if $e_{ij}=Ar_{ij}$, where $a\le i\le j\le n$

Given $A\in Tn(R)$ show thast A is a scalar matrix if $e_{ij}=Ar_{ij}$, where $a\le i\le j\le n$ Prove for $1: e_{ii}A=Ae_{ii}$ and for $2: e_{ij}A=Ae_{ij}$ where $i\le j$ Now for 1, I understand ...