Suppose $A$ is $n \times n$ and the equation $Ax=b$ has a solution for each $b$ in $\Bbb R^n$. Explain why $A$ must be invertible. [Hint: Is $A$ row equivalent to $I_n$]

Explain why the columns of an $n \times n$ matrix $A$ span $\Bbb R^n$ when $A$ is invertible. [Hint: Review Theorem 4 in Section 1.4.]

Suppose $A$, $B$, and $X$ are $n \times n$ matrices with $A$, $X$, and $A-AX$ invertible, and suppose $(A-AX)^{-1}=X^{-1}B$ (eq. 3) a. Explain why $B$ is invertible ...

Suppose $A$ and $B$ are $n \times n$, B is invertible, and AB is invertible. Show that $A$ is invertible [Hint: Let $C=AB$, and solve this equation for $A$

If $u$ and $v$ are in $\Bbb R^n$, how are $u^Tv$ and $v^Tu$ related? How are $uv^T$ and $vu^T$ related?

Suppose $A$ is an $3 \times n$ matrix who columns span $\Bbb R^3$. Explain how to construct an $n \times 3$ matrix $D$ such that $AD=I_3$

A and B are involved in a duel. The rules of the duel are that they are to pick up their guns and shoot at each other simultaneously. If one or both are hit, then the duel is...

Barbara and Dianne go target shooting. Suppose that each of Barbara’s shots hits a wooden duck target with probability p1, while each shot of Dianne’s hits it with...

Independent flips of a coin that lands on heads with probability p are made. What is the probability that the first four outcomes are (a) H, H, H, H? (b) T, H, H, H? ...

Prostate cancer is the most common type of cancer found in males. As an indicator of whether a male has prostate cancer, doctors often perform a test that measures the level of...

There are 3 coins in a box. One is a two-headed coin, another is a fair coin, and the third is a biased coin that comes up heads 75 percent of the time. When one of the 3 coins...

Twelve percent of all U.S. households are in California. A total of 1.3 percent of all U.S. households earn more than \\$250,000 per year, while a total of 3.3 percent of all...

On rainy days, Joe is late to work with probability .3; on nonrainy days, he is late with probability .1. With probability .7, it will rain tomorrow. (a) Find the...

Ms. Aquina has just had a biopsy on a possibly cancerous tumor. Not wanting to spoil a weekend family event, she does not want to hear any bad news in the next few days. But if...

Suppose that 5 percent of men and .25 percent of women are color blind. A color-blind person is chosen at random. What is the probability of this person being male? Assume that...

In a certain community, 36 percent of the families own a dog and 22 percent of the families that own a dog also own a cat. In addition, 30 percent of the families own a cat....

A certain town with a population of 100,000 has 3 newspapers: I, II, and III. The proportions of townspeople who read these papers are as follows: I: 10 percent I and II:...

In an experiment, die is rolled continually until a 6 appears, at which point the experiment stops. What is the sample space of this experiment? Let En denote the event that n...