Question

Consider the three following matrices: A=begin{bmatrix}1 & 0&0 0 & -1&00&0&1 end{bmatrix} , B=begin{bmatrix}1 & 0&0 0 & 0&00&0&-1 end{bmatrix} text{ and } C=begin{bmatrix}0 & -i&0 i & 0&-i0&i&0 end{bmatrix} Calculate the Tr(ABC) (a)1 (b)2 (c)2i (d)0

Matrices
ANSWERED
asked 2020-10-28
Consider the three following matrices:
\(A=\begin{bmatrix}1 & 0&0 \\0 & -1&0\\0&0&1 \end{bmatrix} , B=\begin{bmatrix}1 & 0&0 \\0 & 0&0\\0&0&-1 \end{bmatrix} \text{ and } C=\begin{bmatrix}0 & -i&0 \\i & 0&-i\\0&i&0 \end{bmatrix}\)
Calculate the Tr(ABC)
(a)1
(b)2
(c)2i
(d)0

Answers (1)

2020-10-29
Step 1
The given matrices are
\(A=\begin{bmatrix}1 & 0&0 \\0 & -1&0\\0&0&1 \end{bmatrix} , B=\begin{bmatrix}1 & 0&0 \\0 & 0&0\\0&0&-1 \end{bmatrix} \text{ and } C=\begin{bmatrix}0 & -i&0 \\i & 0&-i\\0&i&0 \end{bmatrix}\)
Step 2
Finding the product ABC.
\(ABC=\begin{bmatrix}1 & 0&0 \\0 & -1&0\\0&0&1 \end{bmatrix}\begin{bmatrix}1 & 0&0 \\0 & 0&0\\0&0&-1 \end{bmatrix}\begin{bmatrix}0 & -i&0 \\i & 0&-i\\0&i&0 \end{bmatrix}\)
\(=\begin{bmatrix}1 & 0&0 \\0 & 0&0\\0&0&-1 \end{bmatrix}\begin{bmatrix}0 & -i&0 \\i & 0&-i\\0&i&0 \end{bmatrix}\)
\(=\begin{bmatrix}0 & -i&0 \\0 & 0&0\\0&-i&0 \end{bmatrix}\)
Hence,
\(ABC=\begin{bmatrix}0 & -i&0 \\0 & 0&0\\0&-i&0 \end{bmatrix}\)
Step 3
Tr(ABC)=sum of the diagonal enteries of the matrix ABC
Tr(ABC)=0+0+0
\(\Rightarrow Tr(ABC)=0\)
Hence, option 'd' is correct.
0
 
Best answer

expert advice

Have a similar question?
We can deal with it in 3 hours

Relevant Questions

asked 2021-05-14
Consider the accompanying data on flexural strength (MPa) for concrete beams of a certain type.
\(\begin{array}{|c|c|}\hline 11.8 & 7.7 & 6.5 & 6 .8& 9.7 & 6.8 & 7.3 \\ \hline 7.9 & 9.7 & 8.7 & 8.1 & 8.5 & 6.3 & 7.0 \\ \hline 7.3 & 7.4 & 5.3 & 9.0 & 8.1 & 11.3 & 6.3 \\ \hline 7.2 & 7.7 & 7.8 & 11.6 & 10.7 & 7.0 \\ \hline \end{array}\)
a) Calculate a point estimate of the mean value of strength for the conceptual population of all beams manufactured in this fashion. \([Hint.\ ?x_{j}=219.5.]\) (Round your answer to three decimal places.)
MPa
State which estimator you used.
\(x\)
\(p?\)
\(\frac{s}{x}\)
\(s\)
\(\tilde{\chi}\)
b) Calculate a point estimate of the strength value that separates the weakest \(50\%\) of all such beams from the strongest \(50\%\).
MPa
State which estimator you used.
\(s\)
\(x\)
\(p?\)
\(\tilde{\chi}\)
\(\frac{s}{x}\)
c) Calculate a point estimate of the population standard deviation ?. \([Hint:\ ?x_{i}2 = 1859.53.]\) (Round your answer to three decimal places.)
MPa
Interpret this point estimate.
This estimate describes the linearity of the data.
This estimate describes the bias of the data.
This estimate describes the spread of the data.
This estimate describes the center of the data.
Which estimator did you use?
\(\tilde{\chi}\)
\(x\)
\(s\)
\(\frac{s}{x}\)
\(p?\)
d) Calculate a point estimate of the proportion of all such beams whose flexural strength exceeds 10 MPa. [Hint: Think of an observation as a "success" if it exceeds 10.] (Round your answer to three decimal places.)
e) Calculate a point estimate of the population coefficient of variation \(\frac{?}{?}\). (Round your answer to four decimal places.)
State which estimator you used.
\(p?\)
\(\tilde{\chi}\)
\(s\)
\(\frac{s}{x}\)
\(x\)
asked 2021-01-22
Given the matrix
\(A=\begin{bmatrix}0 & 0&1 \\ 0 & 3&0 \\ 1&0 & -10 \end{bmatrix}\)
and suppose that we have the following row reduction to its PREF B
\(A=\begin{bmatrix}0 & 0&1 \\ 0 & 3&0 \\ 1&0 & -10 \end{bmatrix}\Rightarrow\begin{bmatrix}1 & 0&-10 \\ 0 & 3&0 \\ 0&0 & 1 \end{bmatrix}\Rightarrow\begin{bmatrix}1 & 0&-10 \\ 0 & 1&0 \\ 0&0 & 1 \end{bmatrix}\Rightarrow\begin{bmatrix}1 & 0&0 \\ 0 & 1&0 \\ 0&0 & 1 \end{bmatrix}\)
Write \(A \text{ and } A^{-1}\) as product of elementary matrices.
asked 2021-02-22

Reduce the following matrices to row echelon form and row reduced echelon forms:
\((i)\begin{bmatrix}1 & p & -1 \\ 2 & 1 & 7 \\ -3 & 3 & 2 \end{bmatrix} (ii) \begin{bmatrix}p & -1 & 7&2 \\ 2 & 1 & -5 & 3 \\ 1 & 3 & 2 & 0 \end{bmatrix}\)
*Find also the ra
of these matrices.
*Notice: \(p=4\)

asked 2020-12-07

Let \(A=\begin{bmatrix}1 & 2 \\-1 & 1 \end{bmatrix} \text{ and } C=\begin{bmatrix}-1 & 1 \\2 & 1 \end{bmatrix}\)
a)Find elementary matrices \(E_1 \text{ and } E_2\) such that \(C=E_2E_1A\)
b)Show that is no elementary matrix E such that \(C=EA\)

...