True or False: For n xx n matrices A and B, define A ox B = AB − BA. The operator ox is not associative or commutative.

Question

True or False: For n xx n matrices A and B, define A ox B = AB − BA. The operator ox is not associative or commutative.

Question

Answers (1)

Relevant Questions

asked 2021-02-05

Let A be nonepty set and P(A) be the power set of A. Recall the definition of power set:
\(\displaystyle{P}{\left({A}\right)}={\left\lbrace{x}{\mid}{x}\subseteq{A}\right\rbrace}\)
Show that symmetric deference operation on P(A) define by the formula
\(\displaystyle{x}\oplus{y}={\left({x}\cap{y}^{{c}}\right)}\cup{\left({y}\cap{x}^{{c}}\right)},{x}\in{P}{\left({A}\right)},{y}\in{p}{\left({A}\right)}\)
(where \(\displaystyle{y}^{{c}}\) is the complement of y) the following statement istrue:
The algebraic operation o+ is commutative and associative on P(A).

See answers (1)

asked 2021-02-03

Given the groups \(R\times\) and Z, let \(G = R×Z\). Define a binary operation \(\circ\) on G by \((a,m)\times(b,n) = (ab,m + n)\). Show that G is a group under this operation.

See answers (1)

asked 2020-11-08

Suppose that a and b belong to a commutative ring and ab is a zero-divisor. Show that either a or b is a zero-divisor.

See answers (1)

asked 2021-05-05

A random sample of \( n_1 = 14 \) winter days in Denver gave a sample mean pollution index \( x_1 = 43 \).
Previous studies show that \( \sigma_1 = 19 \).
For Englewood (a suburb of Denver), a random sample of \( n_2 = 12 \) winter days gave a sample mean pollution index of \( x_2 = 37 \).
Previous studies show that \( \sigma_2 = 13 \).
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
\( H_0:\mu_1=\mu_2.\mu_1>\mu_2 \)
\( H_0:\mu_1<\mu_2.\mu_1=\mu_2 \)
\( H_0:\mu_1=\mu_2.\mu_1<\mu_2 \)
\( H_0:\mu_1=\mu_2.\mu_1\neq\mu_2 \)
(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference \( \mu_1 - \mu_2 \). Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
(e) Based on your answers in parts (i)−(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \alpha?
At the \( \alpha = 0.01 \) level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the \( \alpha = 0.01 \) level, we reject the null hypothesis and conclude the data are statistically significant.
At the \( \alpha = 0.01 \) level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the \( \alpha = 0.01 \) level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
\( \mu_1 - \mu_2 \).
(Round your answers to two decimal places.)
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.

See answers (1)

asked 2020-10-19

Show that a ring is commutative if it has the property that ab = ca implies b = c when \(\displaystyle{a}\ne{0}\).

See answers (1)

asked 2020-12-09

If A and B are ideals of a commutative ring R with unity and A+B=R show that \(\displaystyle{A}\cap{B}={A}{B}\)

See answers (1)

asked 2020-11-10

If \(A=\begin{bmatrix}-2 & 1&-4 \\-2 & 4&-1 \\ 1 &-1 &-4 \end{bmatrix} \text{ and } B=\begin{bmatrix}-2 & 4&2 \\-4 & -1&1 \\ 4 &1 &1 \end{bmatrix}\)
then AB=?
BA=?
True or false : AB=BA for any two square matrices A and B of the same size.

See answers (1)

asked 2021-01-04

In the following question there are statements which are TRUE and statements which are FALSE.
Choose all the statements which are FALSE.
1. If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent - thus no solution.
2. If B has a column with zeros, then AB will also have a column with zeros, if this product is defined.
3. If AB + BA is defined, then A and B are square matrices of the same size/dimension/order.
4. Suppose A is an n x n matrix and assume A^2 = O, where O is the zero matrix. Then A = O.
5. If A and B are n x n matrices such that AB = I, then BA = I, where I is the identity matrix.

See answers (1)

asked 2021-02-19

Determine whether each of the following statements is true or false, and explain why.If A and B are square matrices of the same size, then AB = BA

See answers (1)

asked 2020-12-14

Let R and S be commutative rings. Prove that (a, b) is a zero-divisor in \(\displaystyle{R}\oplus{S}\) if and only if a or b is a zero-divisor or exactly one of a or b is 0.

See answers (1)

Answer 1

Let A, B, C are all in matrices.
Then,
\(\displaystyle{A}\otimes{\left({B}\otimes{C}\right)}={A}\otimes{\left({B}{C}-{C}{B}\right)}\)
\(\displaystyle={A}{\left({B}{C}-{C}{B}\right)}-{\left({B}{C}-{C}{B}\right)}\)
\(\displaystyle={A}{B}{C}-{A}{C}{B}-{B}{C}{A}+{C}{B}{A}\ldots{\left({1}\right)}\)
\(\displaystyle{\left({A}\otimes{B}\right)}\otimes{C}={\left({A}{B}-{B}{A}\right)}\otimes{C}\)
\(\displaystyle={\left({A}{B}-{B}{A}\right)}{C}-{C}{\left({A}{B}-{B}{A}\right)}\)
\(\displaystyle={A}{B}{C}-{B}{A}{C}-{C}{A}{B}+{C}{B}{A}\ldots{\left({2}\right)}\)
Since (1) and (2) are not equal therefore
\(\displaystyle{A}\otimes{\left({B}\otimes{C}\right)}\ne{\left({A}\otimes{B}\right)}\otimes{C}\)
Hence ox id not associative
Now, \(\displaystyle{A}\otimes{B}={A}{B}-{B}{A}\)
\(\displaystyle{B}\otimes{A}={B}{A}-{A}{b}\)
\(\displaystyle=-{\left({A}{B}-{B}{A}\right)}\)
Therefore \(\displaystyle{A}\otimes{B}\ne{B}\otimes{A}\)
Hence \(\displaystyle\otimes\) is not commutative.

Didn’t find what you are looking for?

True or False: For n xx n matrices A and B, define A ox B = AB − BA. The operator ox is not associative or commutative.

True or False: For n xx n matrices A and B, define A ox B = AB − BA. The operator ox is not associative or commutative.

Answers (1)

Relevant Questions