If you toss a coin 6 times and list the results (heads or tails) in order, how many different orders are possible?

Mathematics

1 answer:

goldenfox [79]3 years ago

3 0

The correct answer is 2 times

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Find the characteristic polynomial of A. Use x for the variable in your polynomial.You do not need to factor your polynomial. A

Vesna [10]

Answer: The required characteristic polynomial of the given matrix A is $x^3+6x+11x+6=0.$

Step-by-step explanation: We are given to find the characteristic polynomial of the following 3 × 3 matrix A with unknown variable x :

$A=\left[\begin{array}{ccc}0&0&1\\4&-3&4\\-2&0&-3\end{array}\right].$

We know that

for any square matrix M, the characteristic polynomial is given by

$|M-xI|=0,$ where I is an identity matrix of same order as M.

Therefore, the characteristic polynomial of matrix A is

$|A-xI|=0\\\\\\\Rightarrow \left|\left[\begin{array}{ccc}0&0&1\\4&-3&4\\-2&0&-3\end{array}\right]-x\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right]\right|=0\\\\\\\Rightarrow \left|\left[\begin{array}{ccc}-x&0&1\\4&-3-x&4\\-2&0&-3-x\end{array}\right] \right|=0\\\\\\\Rightarrow -x(3+x)^2+1(0-6-2x)=0\\\\\Rightarrow (x+3)(-3x-x^2-2)=0\\\\\Rightarrow (x+3)(x^2+3x+2)=0\\\\\Rightarrow x^3+6x+11x+6=0.$

Thus, the required characteristic polynomial of the given matrix A is $x^3+6x+11x+6=0.$

6 0

3 years ago

Help

Art [367]

$\huge\fbox{Answer ☘}$

$\bold{3( \frac{27}{8} ) {}^{ \frac{2}{3} \times - 1} = 2( \frac{32}{243} ) {}^{2x} }\\ \\3(( \frac{3}{2} ) {}^{3} ) {}^{ \frac{2}{3} \times - 1 } = 2(( \frac{2}{3} ) {}^{5} ) {}^{2x} \\\\ 3( \frac{3}{2} ) {}^{2 \times - 1} = 2( \frac{2}{3} ) {}^{10x} \\\\ 3( \frac{2}{3} ) {}^{2} = 2( \frac{2}{3} ) {}^{10x} \\\\ 3( \frac{4}{9} ) = 2( \frac{4}{9} ) {}^{5x} \\\\\bold\pink{ comparing \: powers }\\\\5x = 1 \\\\ \bold\blue{x = \frac{1}{5} }$