Linear Algebra Exercises

Exercise 1 Consider the homogeneous linear system

\[ \sum_{j=1}^{n} a_jx_j+bx_i=0, \qquad i=1,\ldots,n, \]

where \(\sum_{i=1}^n a_i\ne0\). Determine the conditions on \(a_1,\ldots,a_n\) and \(b\) under which:

1. The system has only the zero solution.
2. The system has nonzero solutions. In this case, determine its solution set.

Exercise 2 Let \(A\) and \(B\) be \(n\times n\) matrices. Prove both bounds:

\[ \begin{aligned} \operatorname{rank}(L_{AB}) &\ge \operatorname{rank}(L_A)+\operatorname{rank}(L_B)-n,\\ \operatorname{rank}(L_{AB}) &\le \min\{\operatorname{rank}(L_A),\operatorname{rank}(L_B)\}. \end{aligned} \]

Exercise 3 Let \(A\) and \(B\) be \(n\times m\) matrices. Prove

\[ \operatorname{rank}(L_{A+B}) \le \operatorname{rank}(L_A)+\operatorname{rank}(L_B). \]

Exercise 4 Let \(A,B\in M_{n\times n}(\mathbb R)\). Suppose \(AB=BA=0\) and \(\operatorname{rank}(A^2)=\operatorname{rank}(A)\). Prove

\[ \operatorname{rank}(A+B)=\operatorname{rank}(A)+\operatorname{rank}(B). \]

Exercise 5 Let \(A,B,C,D\) be pairwise commuting \(n\times n\) matrices satisfying \(AC+BD=I_n\). Prove

\[ \operatorname{rank}(AB)=\operatorname{rank}(A)+\operatorname{rank}(B)-n. \]

Exercise 6 Let \(A\) be an \(n\times n\) real matrix satisfying \(AA^t=k^2I_n\), where \(k\in\mathbb R\). Prove

\[ \operatorname{rank}(kI_n-A)=\operatorname{rank}\big((kI_n-A)^2\big). \]

Exercise 7 Let \(A\) be an \(n\times n\) matrix with rank \(r\), where \(1\le r\le n/2\), and suppose \(A^2=0\). Prove that an invertible matrix \(P\) exists such that

\[ A=P \begin{pmatrix} 0&I_r&0\\ 0&0&0\\ 0&0&0 \end{pmatrix} P^{-1}, \]

where \(I_r\) is the \(r\times r\) identity matrix and each \(0\) denotes a zero matrix of the appropriate size.

Exercise 8 Let \(A\) be a real square matrix satisfying \(A^2=I_n\). Prove that \(A\) is diagonalizable.

Exercise 9 Let \(V\) be a finite-dimensional real inner product space, and let \(U\) and \(T\) be self-adjoint linear operators on \(V\) satisfying \(UT=TU\).

Prove that \(V\) has an orthonormal basis consisting of common eigenvectors of \(U\) and \(T\).

Exercise 10 Let \(A\) and \(B\) be \(n\times n\) complex matrices satisfying \(AB=A+B\).

Prove that \(A\) and \(B\) are simultaneously triangularizable: there is a basis in which both matrices are upper triangular.

Exercise 11 Let \(A\) be an \(n\times n\) matrix satisfying

\[ \sum_{i=0}^{m}b_iA^i=0, \qquad |b_m|>\sum_{i=0}^{m-1}|b_i|. \]

Prove that the equation \(2X-XA^2=AX\) has only the trivial solution \(X=0\).