Let U=begin{bmatrix}frac{sqrt2}{2} & frac{sqrt3}{3} -frac{sqrt2}{2} & frac{sqrt3}{3} 0&-frac{sqrt3}{3} end{bmatrix} text{ and } x=begin{bmatrix}1 2 end{bmatrix} b)Compute | Ux | (it should be the same as | x |)

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Let U=begin{bmatrix}frac{sqrt2}{2} & frac{sqrt3}{3} -frac{sqrt2}{2} & frac{sqrt3}{3} 0&-frac{sqrt3}{3} end{bmatrix} text{ and } x=begin{bmatrix}1 2 end{bmatrix} b)Compute | Ux | (it should be the same as | x |)

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asked 2021-02-24

Let \(u=\begin{bmatrix}2 \\ 5 \\ -1 \end{bmatrix} , v=\begin{bmatrix}4 \\ 1 \\ 3 \end{bmatrix} \text{ and } w=\begin{bmatrix}-4 \\ 17 \\ -13 \end{bmatrix}\) It can be shown that \(4u-3v-w=0\). Use this fact (and no row operations) to find a solution to the system \(4u-3v-w=0\) , where
\(A=\begin{bmatrix}2 & -4 \\5 & 17\\-1&-13 \end{bmatrix} , x=\begin{bmatrix}x_1 \\ x_2 \end{bmatrix} , b=\begin{bmatrix}4 \\ 1 \\ 3 \end{bmatrix}\)

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asked 2020-10-25

Solve for X in the equation, given
\(3X + 2A = B\)
\(A=\begin{bmatrix}-4 & 0 \\1 & -5\\-3&2 \end{bmatrix} \text{ and } B=\begin{bmatrix}1 & 2 \\ -2 & 1 \\ 4&4 \end{bmatrix}\)

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asked 2020-11-30

Let M be the vector space of \(2 \times 2\) real-valued matrices.
\(M=\begin{bmatrix}a & b \\c & d \end{bmatrix}\)
and define \(M^{\#}=\begin{bmatrix}d & b \\c & a \end{bmatrix}\) Characterize the matrices M such that \(M^{\#}=M^{-1}\)

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asked 2020-10-27

Let \(A=\begin{bmatrix}2 & -1&5 \\-3 & 4&0 \end{bmatrix} \text{ and } B=\begin{bmatrix}-3 & -4&2 \\-1 & 0&-5 \end{bmatrix}\)
Find each result.
3A+2B=?
A-3B=?

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asked 2021-02-09

Compute the indicated matrices, if possible .
A^2B
let \(A=\begin{bmatrix}1 & 2 \\3 & 5 \end{bmatrix} \text{ and } B=\begin{bmatrix}2 & 0 & -1 \\3 & -3 & 4 \end{bmatrix}\)

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asked 2021-02-06

Let the matrix A,B, C and D be as given. Find the product of the sum of A and B and the difference between C and D.
\(A=\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix},B=\begin{bmatrix}1 & 0 \\0 & -1 \end{bmatrix},C=\begin{bmatrix}-1 & 0 \\0& 1 \end{bmatrix},D=\begin{bmatrix}-1 & 0 \\0 & -1 \end{bmatrix}\)

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asked 2020-11-08

Classify each of the following matrices according as it is (a) real, (b) symmetric, (c) skew-symmetric, (d) Hermitian, or (e) skew-hermitian, and identify its principal and secondary diagonals.
\(\begin{bmatrix}1 & 0&-i \\ 0 & -2 & 4-i \\ i&4+i&3 \end{bmatrix}\)
\(\begin{bmatrix}7 & 0&4 \\ 0 & -2 & 10 \\ 4&10&5 \end{bmatrix}\)

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asked 2021-02-05

\(A=\begin{bmatrix}2& 1&1 \\-1 & -1&4 \end{bmatrix} B=\begin{bmatrix}0& 2 \\-4 & 1\\2 & -3 \end{bmatrix} C=\begin{bmatrix}6& -1 \\3 & 0\\-2 & 5 \end{bmatrix} D=\begin{bmatrix}2& -3&4 \\-3 & 1&-2 \end{bmatrix}\)
a)\(A-3D\)
b)\(B+\frac{1}{2}\)
c) \(C+ \frac{1}{2}B\)
(a),(b),(c) need to be solved

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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.

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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

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Answer 1

Step 1
Given matrices are,
\(U=\begin{bmatrix}\frac{\sqrt2}{2} & \frac{\sqrt3}{3} \\-\frac{\sqrt2}{2} & \frac{\sqrt3}{3} \\0&-\frac{\sqrt3}{3} \end{bmatrix} \text{ and } x=\begin{bmatrix}1 \\2 \end{bmatrix}\)
Step 2 Now the multiplication of the matrices is,
\(Ux=\begin{bmatrix}\frac{\sqrt2}{2} & \frac{\sqrt3}{3} \\-\frac{\sqrt2}{2} & \frac{\sqrt3}{3} \\0&-\frac{\sqrt3}{3} \end{bmatrix}\begin{bmatrix}1 \\2 \end{bmatrix}=\begin{bmatrix}\frac{\sqrt2}{2}+ \frac{2\sqrt3}{3} \\-\frac{\sqrt2}{2}+ \frac{2\sqrt3}{3} \\-\frac{2\sqrt3}{3} \end{bmatrix}\)
Step 3
Norm of the matrix is,
\|Ux\|=\begin{Vmatrix}\begin{bmatrix}\frac{\sqrt2}{2}+ \frac{2\sqrt3}{3} \\-\frac{\sqrt2}{2}+ \frac{2\sqrt3}{3} \\-\frac{2\sqrt3}{3} \end{bmatrix}\end{Vmatrix}=\sqrt{\left(\frac{\sqrt2}{2}+ \frac{2\sqrt3}{3}\right)^2+\left(-\frac{\sqrt2}{2}+ \frac{2\sqrt3}{3}\right)^2+\left(-\frac{2\sqrt3}{3}\right)^2}\)
=\sqrt{\frac{2}{4}+\frac{12}{9}+2\frac{\sqrt2}{2}\frac{2\sqrt3}{2}+\frac{2}{4}+\frac{12}{9}-2\frac{\sqrt2}{2}\frac{2\sqrt3}{2}+\frac{12}{9}}\)
=\sqrt{\frac{1}{2}+\frac{4}{3}+\frac{1}{2}+\frac{4}{3}+\frac{4}{3}}=\sqrt{1+\frac{12}{3}}=\sqrt{5}\)
Step 4
Therefore, norm of the matrix is,
\(\|Ux\|=\sqrt{5}\)

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Let U=begin{bmatrix}frac{sqrt2}{2} & frac{sqrt3}{3} -frac{sqrt2}{2} & frac{sqrt3}{3} 0&-frac{sqrt3}{3} end{bmatrix} text{ and } x=begin{bmatrix}1 2 end{bmatrix} b)Compute | Ux | (it should be the same as | x |)

Let U=begin{bmatrix}frac{sqrt2}{2} & frac{sqrt3}{3} -frac{sqrt2}{2} & frac{sqrt3}{3} 0&-frac{sqrt3}{3} end{bmatrix} text{ and } x=begin{bmatrix}1 2 end{bmatrix} b)Compute | Ux | (it should be the same as | x |)

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