### Questions (16)

 7 We have matrix $A\in M_{n-1\times n}(\mathbb Z)$ so that the sum of entries in each row is zero. Prove that $\det(AA^T)=nk^2.$ 6 Prove that, at least one of the matrices $A+B$ and $A-B$ has to be singular 4 Prove that there is a base of $\mathbb R^4$ made of eigenvectors of matrix $A$ 4 Let $A$=[$a_{ij}$]$_{n x n}$ where $a_{ii}$=$1$, $i=\overline {1,n}$, $a_{ij}=a\not=1, i\not=j$. Find $A^n$, $n\in \mathbb N$ 3 $\{v_1,,...,v_{2014}\}$ are linearly independent. Find $\lambda$ so that $\{v_1+v_2,...,v_{2014}+\lambda v_1\}$ are also linearly independent.

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### Tags (11)

 0 linear-algebra × 14 0 vector-spaces × 2 0 matrices × 7 0 magma 0 eigenvalues-eigenvectors × 3 0 polynomials 0 complex-numbers × 2 0 vectors 0 linear-transformations × 2 0 abstract-algebra

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 11 Is sum of two orthogonal matrices singular? 4 Show that sum of elements of rows / columns of a matrix is equal to reciprocal of sum of elements of rows/colums of its inverse matrix 4 Let $\mathcal L$, $\mathcal G$ : $V$ $\rightarrow$ $V$ be two linear operators, prove the following 2 Determine basis of matrix 1 Find $t$ such that (subspace)