# Matrices Solved Problems

(Such a matrix is an example of a nilpotent matrix.

Then prove that $\R^n=\im(T) \oplus \ker(T).$ Read solution Suppose that $A$ is $2\times 2$ matrix that has eigenvalues $-1$ and $3$.

Then for each positive integer $n$ find $a_n$ and $b_n$ such that $A^=a_n A b_n I,$ where $I$ is the $2\times 2$ identity matrix.

Prove that $\$ is a basis of $U$ and conclude that the dimension of $U$ is $2$.

(b) Let $T$ be a map from $U$ to $U$ defined by $T\big((a_1, a_2, \dots)\big)=(a_2, a_3, \dots). (b) Assume that A is an n\times n nonzero idempotent matrix. Then determine all integers k such that the matrix I-k A is idempotent. (b) Use the result of (a), find a sequence (a_i)_^ satisfying a_1=2, a_2=7. Read solution Let A, B, and C be n \times n matrices and I be the n\times n identity matrix. (a) If A is similar to B, then B is similar to A. (c) If A is similar to B and B is similar to C, then A is similar to C. (d) If A is similar to the identity matrix I, then A=I. Read solution (a) Let A=\begin 1 & 2 & 1 \ 3 &6 &4 \end and let \[\mathbf=\begin -3 \ 1 \ 1 \end, \qquad \mathbf=\begin -2 \ 1 \ 0 \end, \qquad \mathbf=\begin 1 \ 1 \end.$ For each of the vectors $\mathbf, \mathbf, \mathbf$, determine whether the vector is in the null space $\cal N(A)$. (b) Find a basis of the null space of the matrix $B=\begin 1 & 1 & 2 \ -2 &-2 &-4 \end$.

(e) If $A$ or $B$ is nonsingular, then $AB$ is similar to $BA$. Read solution Let $V$ be a real vector space of all real sequences $(a_i)_^=(a_1, a_2, \dots).$ Let $U$ be the subspace of $V$ consisting of all real sequences that satisfy the linear recurrence relation $a_-5a_ 3a_=0$ for $k=1, 2, \dots$.

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