Let $N$ be an $n \times n$ matrix with coefficients in the field $F .$ The matrix $N$ is said to be nilpotent if some power of $N$ is the zero matrix, i.e., $N^{k}=0$ for some $k .$ Prove that any nilpotent matrix is similar to a block diagonal matrix whose blocks are matrices with 1 's along the first superdiagonal and 0 's elsewhere.

Find slope-intercept form of the equation of the line that passes through the given points P and Q. P = (6, 2) and Q = (7, -1)

Write the express ion for the solubility-product constant for each of the following ionic compounds:

$$SrSO_4$$

Find the derivative of y with respect to the given independent variable:$y=\log _{3}\left(\left(\frac{x+1}{x-1}\right)^{\ln 3}\right)$

What kinds of processes require the use of diamond tools?

Rewrite the following dialog, adding punctuation, capitalization, indentation, and speaker tags as needed. - You're kidding! Neil said. Marisa explained we had to read it in class.

Cara's muffin recipe calls for

$$1\frac{1}{2}$$

cups of flour for the muffins and

$$\frac{1}{4}$$

cup of flour for the topping. If she makes

$$\frac{1}{2}$$

of the original recipe, how much flour will she use?

For matrix

$$\begin{bmatrix}2&4\\-1&-2\end{bmatrix}$$

compute its eigenvalues and eigenvector(s), and sketch the eigenspace if the eigenvectors are real.

The numerator of a fraction is five less than the denominator. The sum of the fraction and six times its reciprocal is $\frac { 25 } { 2 } .$ Find the numerator and denominator assuming that each is an integer.

What is the difference between adaptation and evolution?

Let $R$ be a P.I.D. let $B$ be a torsion $R$ -module and let $p$ be a prime in $R .$ Prove that if $p b=0$ for some nonzero $b \in B,$ then $\operatorname{Ann}(B) \subseteq(p)$

Let $\lambda$ be an eigenvalue of the linear transformation $T$ on the finite dimensional vectorspace $V$ over the field $F .$ Let $r_{k}=\operatorname{dim}_{F}(T-\lambda)^{k} V$ be the rank of the linear transformation $(T-\lambda)^{k}$ on $V .$ For any $k \geq 1,$ prove that $r_{k-1}-2 r_{k}+r_{k+1}$ is the number of Jordan blocks of $T$ corresponding to $\lambda$ of size $k[u s e$ Exercise 12 in Section 1$] .$ (This gives an efficient method for determining the Jordan canonical form for $T$ by computing the ranks of the matrices $(A-\lambda I)^{k}$ for a matrix $A$ representing $T,$ cf. Exercise 31$($ a) in Section $11.2 .)$

Use the method of Example 2 above to determine explicit matrices $P_{1}$ and $P_{2}$ with $P_{1}^{-1} B P_{1}$ and $P_{2}^{-1} C P_{2}$ in Jordan canonical form. Use this to explicitly construct a matrix $Q$ which conjugates $B$ into $C$ (proving directly that these matrices are similar).

Let $R$ be an integral domain and let $A$ and $B$ be $R$ -modules of ranks $m$ and $n,$ respectively. Prove that the rank of $A \oplus B$ is $m+n .$ [Use the previous exercise.]