*(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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