Solve the matrix equation for X. [1 -1 1, 2 3 0, 0 2 -1] X = [2 - 1 5 7 8, 4 0 -3, 0 1, 3 5 -7 2 1]

Write the number in scientific notation. −5,222,145

Determine conditions on the b1's, if any, in order to guarantee that the linear system is consistent. x1 - 2x2 - x3 = b1, -4x1 + 5x2 + 2x3 = b2, -4x1 + 7x2 + 4x3 = b3

Find all values of c, if any, for which the given matrix is invertible. [c 1 0, 1 c 1, 0 1 c]

Prove: If A and B are n x n matrices, then tr(A + B ) = tr(A) + tr(B)

Find the time required, to the nearest 0.1 year, for the investment to reach the desired goal. $2,000 invested at 6.75%, compounded daily; goal:$3,000.

Find all values of c, if any, for which the given matrix is invertible. [c c c, 1 c c, 1 1 c]

Determine conditions on the b1's, if any, in order to guarantee that the linear system is consistent. 6x1 - 4x2 = b1, 3x1 - 2x2 = b2

Which compound gives a molecular ion at m/z = 122:

$$\mathrm { C } _ { 6 } \mathrm { H } _ { 5 } \mathrm { CH } _ { 2 } \mathrm { CH } _ { 2 } \mathrm { CH } _ { 3 }, \mathrm { C } _ { 6 } \mathrm { H } _ { 5 } \mathrm { COCH } _ { 2 } \mathrm { CH } _ { 3 } , \text { or } \mathrm { C } _ { 6 } \mathrm { H } _ { 5 } \mathrm { OCH } _ { 2 } \mathrm { CH } _ { 3 } ?$$

If A and B are invertible matrices, show that AB and BA are similar.

Solve the linear systems together by reducing the appropriate augmented matrix. x1 + 3x2 + 5x3 = b1, -x1 - 2x2 = b2, 2x1 + 5x2 + 4x3 = b3, (i) b1 = 1, b2 = 0, b3 = -1, (ii) b1 = 0, b2 = 1, b3 = 1, (iii) b1 = -1, b2 = -1, b3 = 0

Write each set in set-builder notation.

$$M = \{ 1,3,5,7,9,11,13,17,19 \}$$

A vitreous silica optical fiber of diameter $100 \mu \mathrm{m}$ is used to send optical signals from a sensor placed deep inside a hydrogen chamber. The hydrogen is at a pressure of 20 bars. The mass diffusivity and solubility of the hydrogen in the glass fiber are $D_{\mathrm{AB}}=2.88 \times 10^{-15}\mathrm{m}^{2} / \mathrm{s}$ and $S=4.15 \times 10^{-3} \mathrm{kmol} / \mathrm{m}^{3} \cdot \mathrm { bar }$, respectively. Hydrogen diffusion into the fiber is undesirable, since it changes the spectral transmissivity and refractive index of the glass and can lead to failure of the detection system. (a) Determine the average hydrogen concentration in an uncoated optical fiber, $\bar{C}$, after 100 h of operation in the hydrogen environment. Determine the corresponding change in the refractive index, $\Delta n$, of the fiber. For vitreous silica, $\Delta n=\left(1.6 \times 10^{-3} \mathrm{m}^{3} / \mathrm{kmol}\right) \times \bar{C}$. (b) Determine the average hydrogen concentration and change in refractive index after 1 h and 10 h of operation in the hydrogen environment.

r(t) is the position vector of a moving particle. Graph the curve and the velocity and acceleration vectors at the indicated time. Find the speed at that time. $\mathbf{r}(t)=-\cosh 2 t \mathbf{i}+\sinh 2 t \mathbf{j} ; t=0$