Let A and B be $n \times n$ matrices, and suppose that B and AB are both invertible. Prove that A is also invertible.

Let A be an invertible $n \times n$ matrix, and let B be an $n \times p$ matrix. Show that the equation AX=B has a unique solution $A^{-1} B$.

Metabolism of arginine produces urea and the rare amino acid ornithine. Ornithine has an isoelectric point close to 10. Propose a structure for ornithine.

Suppose (B – C)D = 0, where B and C are m × n matrices and D is invertible. Show that B = C.

Indicate which of the following choices best identifies the hypothesis test. a. independent group means, population standard deviations and/or variances known b. independent group means, population standard deviations and/or variances unknown c. matched or paired samples d. single mean e. two proportions f. single proportion. The mean number of English courses taken in a two–year time period by male and female college students is believed to be about the same. An experiment is conducted and data are collected from nine males and 16 females.

Find an example that meets the given specifications. A linear transformation $T: \mathrm{R}^{2} \rightarrow \mathrm{R}^{2}$ such that

$$T\left(\left[\begin{array}{l} 0 \\ 1 \end{array}\right]\right)=\left[\begin{array}{l} 1 \\ 4 \\ 5 \end{array}\right]$$

Prove that the given function is a linear transformation. $T: C[0,1] \rightarrow \mathbf{R}^{2}$ with

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

Suppose AB = AC; where B and C are n × p matrices and A is invertible. Show that B = C. Is this true, in general, when A is not invertible?

Suppose CA = In (the n × n identity matrix). Show that the equation Ax = 0 has only the trivial solution. Explain why A cannot have more columns than rows.

Prove that the given function is a linear transformation. $T: V \rightarrow W$ with $T(\mathbf{v})=\mathbf{0}_{W}$ for all v.

Simplify each expression. 5a - 2 - 3a + 7

Suppose P is invertible and $A=P B P^{-1}$. Solve for B in terms of A.

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.