This important concept denotes a matrix that commutes with its conjugate transpose A^A^^T=^A^^TA. Prove that Hermitian, skew-Hermitian, and unitary matrices are normal. Give corresponding examples of your own.

A $\mathrm{CO}_{2}$ cartridge is used to propel a small rocket cart. Compressed gas, stored at 35 MPa and $20^{\circ} \mathrm{C}$, is expanded through a smoothly contoured converging nozzle with 0.5 mm throat diameter. The back pressure is atmospheric. Calculate the pressure at the nozzle throat. Evaluate the mass flow rate of carbon dioxide through the nozzle. Determine the thrust available to propel the cart. How much would the thrust increase if a diverging section were added to the nozzle to expand the gas to atmospheric pressure? What is the exit area? Show stagnation states, static states, and the processes on a Ts diagram.

A theoretical justification based on a certain material failure mechanism underlies the assumption that ductile strength X of a material has a lognormal distribution. Suppose the parameters are $\mu=5$ and $\sigma=.1$. a. Compute E(X) and V(X). b. Compute P(X>125). c. Compute $P(110 \leq X \leq 125)$. d. What is the value of median ductile strength? e. If ten different samples of an alloy steel of this type were subjected to a strength test, how many would you expect to have strength of at least 125? f. If the smallest 5% of strength values were unacceptable, what would the minimum acceptable strength be?