Find a formula for the quadratic form with the given matrix A.

$$A=\left[\begin{array}{llll} 5 & 0 & 1 & 0 \\ 0 & 2 & 0 & 0 \\ 1 & 0 & 0 & 3 \\ 0 & 0 & 3 & 4 \end{array}\right]$$

The lipid(s) used as the basis of vitamin D, sex hormones, and bile salts is/are (a) triglycerides, (b) cholesterol, (c) phospholipids, (d) prostaglandin.

In the following exercise, compute the steady-state matrix of each stochastic matrix.

$$\left[\begin{array}{lllll}1 & 0 & 0 & .2 & .1 \\ 0 & 1 & 0 & .1 & .2 \\ 0 & 0 & 1 & .3 & .1 \\ 0 & 0 & 0 & .2 & .2 \\ 0 & 0 & 0 & .2 & .4\end{array}\right]$$

Given that the augmented matrix in row reduced form is equivalent to the augmented matrix of a system of linear equations, (a) determine whether the system has a solution and (b) find the solution or solutions to the system, if they exist.

$$\left[\begin{array}{rrrr|r} 0 & 1 & 0 & 1 & 3 \\ 0 & 0 & 1 & -2 & 4 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$$

Let the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row-echelon form. Solve the system.

$$\left[\begin{array}{rrrrrr}1 & -6 & 0 & 0 & 3 & -2 \\ 0 & 0 & 1 & 0 & 4 & 7 \\ 0 & 0 & 0 & 1 & 5 & 8 \\ 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]$$

Use the QR Algorithm to determine, to within $10^{−5},$ all the eigenvalues of the following matrices.

$$a. \left[\begin{array}{rrr}{2} & {-1} & {0} \\ {-1} & {-1} & {-2} \\ {0} & {-2} & {3}\end{array}\right]$$

$$b. \left[\begin{array}{lll}{3} & {1} & {0} \\ {1} & {4} & {2} \\ {0} & {2} & {3}\end{array}\right]$$

$$c. \left[\begin{array}{lllll}{4} & {2} & {0} & {0} & {0} \\ {2} & {4} & {2} & {0} & {0} \\ {0} & {2} & {4} & {2} & {0} \\ {0} & {0} & {2} & {4} & {2} \\ {0} & {0} & {0} & {2} & {4}\end{array}\right]$$

$$d. \left[\begin{array}{rrrrr}{5} & {-1} & {0} & {0} & {0} \\ {-1} & {4.5} & {0.2} & {0} & {0} \\ {0} & {0.2} & {1} & {-0.4} & {0} \\ {0} & {0} & {-0.4} & {3} & {1} \\ {0} & {0} & {0} & {1} & {3}\end{array}\right]$$

For each given set of basis vectors, use the Gram-Schmidt method to find an orthonormal set. (a) A = (0, 2, 0, 0), B = (3,−4, 0, 0), C = (1, 2, 3, 4). (b) A = (0, 0, 0, 7), B = (2, 0, 0, 5), C = (3, 1, 1, 4). (c) A = (6, 0, 0, 0), B = (1, 0, 2, 0), C = (4, 1, 9, 2).

For each of the following elementary matrices, describe the corresponding elementary row operation and write the inverse.

$$\text { (a) } E = \left[ \begin{array} { l l l } { 1 } & { 0 } & { 3 } \\ { 0 } & { 1 } & { 0 } \\ { 0 } & { 0 } & { 1 } \end{array} \right]$$

,

$$\text { (b) } E = \left[ \begin{array} { c c c } { 0 } & { 0 } & { 1 } \\ { 0 } & { 1 } & { 0 } \\ { 1 } & { 0 } & { 0 } \end{array} \right]$$

,

$$( \mathrm { c } ) E = \left[ \begin{array} { l l l } { 1 } & { 0 } & { 0 } \\ { 0 } & { \frac { 1 } { 2 } } & { 0 } \\ { 0 } & { 0 } & { 1 } \end{array} \right]$$

,

$$\text { (d) } E = \left[ \begin{array} { r r r } { 1 } & { 0 } & { 0 } \\ { - 2 } & { 1 } & { 0 } \\ { 0 } & { 0 } & { 1 } \end{array} \right]$$

,

$$\text { (e) } E = \left[ \begin{array} { l l l } { 0 } & { 1 } & { 0 } \\ { 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 1 } \end{array} \right]$$

,

$$( f ) E = \left[ \begin{array} { l l l } { 1 } & { 0 } & { 0 } \\ { 0 } & { 1 } & { 0 } \\ { 0 } & { 0 } & { 5 } \end{array} \right]$$

Solve the given problems.
If two objects collide and the kinetic energy remains constant, the collision is termed perfectly elastic. Under these conditions, if an object of mass $m_1$ and initial velocity $u_1$ strikes a second object (initially at rest) of mass $m_2$, such that the velocities after collision are $v_1$ and $v_2$, the following equations are found:

$$\begin{aligned} m_1 u_1 & =m_1 v_1+m_2 v_2 \\ \frac{1}{2} m_1 u_1^2 & =\frac{1}{2} m_1 v_1^2+\frac{1}{2} m_2 v_2^2 \end{aligned}$$

Solve these equations for $m_2$ in terms of $u_1, v_1$, and $m_1$.

Sketch the graph of a function that satisfies all of the given conditions. Vertical asymptote x=0, f'(x)>0 if x<-2, f'(x)<0 if x>-2 (x is not equal to 0), f''(x)<0 if x<0, f''(x)>0 if x>=

In given exercise, find A B, if possible.

$$A=\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & -2\end{array}\right], \quad B=\left[\begin{array}{rrr}3 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 5\end{array}\right]$$

Find a basis for each of the four subspaces of

$$A = \begin{bmatrix} 0 & 1 & 2 & 3.& 4\\ 0 & 1 & 2 & 4 & 6\\ 0 & 0 & 0 & 1 & 2 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0\\ 1 & 1 & 0\\ 0 & 1 & 1 \end{bmatrix} \begin{bmatrix} 0 & 1 & 2 & 3.& 4\\ 0 & 0 & 0 & 1 & 2\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}.$$

You are given an augmented matrix in reduced row echelon form. Write the system (in terms of $x_1, x_2,... , x_n$) represented by the augmented matrix; then write the solution to the system in parametric form.

$$\left( \begin{array}{rrrrrr}{1} & {-1} & {0} & {0} & {1} & {1} \\ {0} & {0} & {1} & {-1} & {-1} & {1} \\ {0} & {0} & {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} & {0} & {0}\end{array}\right)$$

Sketch the graph of a function having the given properties:

• $f(-1) = 0, f'(-1) = 0$
• $f(0) = 1, f'(0) = 0$
• $f'(x) < 0$ on $(-\infin, -1)$
• $f'(x) > 0$ on $(-1, \infin)$
• $f''(x) > 0$ on $(-\infin, -\frac{2}{3}) \cup (0, \infin)$
• $f''(x) < 0$ on $(-\frac{2}{3}, 0)$