Let a matrix A- 1 0- 3 -1- - then A15 can be

Given : A=[ 1 amp;0; 3 amp; 1; ] ⇒ A^2=[ 1 amp;0; 3 amp; 1; ][ 1 amp;0; 3 amp; 1; ]=[ 1 amp;0; 0 a ...

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Diagonal Matrix

Let a matrix ...

Question

Let a matrix $A = [\begin{matrix} 1 & 0 3 & - 1 \end{matrix}],$ then $A^{15}$ can be

A

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A^{2}

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A^{3}

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A^{4}

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Solution

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α

⎡ ⎢ ⎣ \begin{matrix} 1 & α & α^{2} α & 1 & α α^{2} & α & 1 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 - 1 1 \end{matrix} ⎤ ⎥ ⎦

1 + α + α^{2} =

Let $α$ be a root of the equation $x^{2} + x + 1 = 0$ and the matrix

$A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 1 & α & α^{2} 1 & α^{2} & α^{4} \end{matrix} ⎤ ⎥ ⎦$ then the matrix $A^{31}$ is equal to

a_{1}, a_{2}, a_{3}, \dots, a_{15}

a_{1} + a_{2} + \dots + a_{10} = A

a_{6} + a_{7} + \dots + a_{15} = B

B - A = 200

B + A = 860,

a_{15}

P = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 4 & 1 & 0 16 & 4 & 1 \end{matrix} ⎤ ⎥ ⎦

Q = [q_{i j}]

P^{50} - Q = I

\frac{q_{31} + q_{32}}{q_{21}}

ω \neq 1

S

⎡ ⎢ ⎣ \begin{matrix} 1 & a & b ω & 1 & c ω^{2} & ω & 1 \end{matrix} ⎤ ⎥ ⎦

a, b

c

ω

ω^{2}

S

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