Asked by 1 day ago
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 $I_n$]

1 answers and 2 views


Asked by 1 day ago
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.]

1 answers and 2 views


Asked by 1 day ago
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 ...

1 answers and 2 views


Asked by 1 day ago
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 $A$

1 answers and 2 views


Asked by 1 day ago
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?

1 answers and 4 views


Asked by 1 day ago
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 $AD=I_3$

1 answers and 4 views


Asked by 3 days ago
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...

1 answers and 6 views


Asked by 3 days ago
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...

1 answers and 3 views


Asked by 3 days ago
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? ...

1 answers and 3 views


Asked by 3 days ago
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...

1 answers and 3 views


Asked by 3 days ago
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...

1 answers and 3 views


Asked by 3 days ago
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...

1 answers and 7 views


Asked by 3 days ago
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...

1 answers and 7 views


Asked by 3 days ago
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...

1 answers and 4 views


Asked by 3 days ago
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...

1 answers and 6 views


Asked by 3 days ago
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....

1 answers and 6 views


Asked by 8 days ago
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:...

1 answers and 35 views


Asked by 8 days ago
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...

1 answers and 9 views


Asked by 8 days ago
A box contains 3 marbles: 1 red, 1 green, and 1 blue. Consider an experiment that consists of taking 1 marble from the box and then replacing it in the box and drawing a second...

1 answers and 3 views


Asked by 14 days ago
If 12 people are to be divided into 3 committees of respective sizes 3, 4, and 5, how many divisions are possible?

1 answers and 7 views