A is a 2×2 matrix with elgenvectors v1=[1−1] and v2=[11] corresponding to eigenvalues λ1=12 and λ2=2, respectively,and x=[71],Find A8x

Step 1GIven:V1=(1−1); V2=(11) and λ1=12; λ2=2respectivelyAlso given that:X=(71)To find:A8x=?Step 2To obtain: A8=?So,A=(11−11)⇒A2=(11−11)(11−11)=(02−20)=2(01−10)A4=A2×A2=2(01−10)×2(01−10)=4(01−10)×(01−10)=4(−100−1)Step 3A8=A4×A4=4(−100−1)×4(−100−1)=16(−100−1)×(−100−1)=16=16(1001)=16IStep 4To obtain:A8X=?A8X=16(1001)(71)=16(71)=(11216)Step 5Result:A8X=(11216)

Answered: "A is a 2×2 matrix with elgenvectors v1=[1−1]..."

klancimarsn

Answered question

2022-03-01

A is a

2 \times 2

matrix with elgenvectors

v_{1} = [\begin{matrix} 1 \\ - 1 \end{matrix}]

and

v_{2} = [\begin{matrix} 1 \\ 1 \end{matrix}]

corresponding to eigenvalues

λ_{1} = \frac{1}{2}

and

λ_{2} = 2

, respectively,and

x = [\begin{matrix} 7 \\ 1 \end{matrix}]

,
Find

A^{8} x

Answer & Explanation

Warenberg56i

Beginner2022-03-02Added 8 answers

Step 1
GIven: $V_{1} = (\begin{matrix} 1 \\ - 1 \end{matrix})$ ; $V_{2} = (\begin{matrix} 1 \\ 1 \end{matrix})$ and $λ_{1} = \frac{1}{2}$ ; $λ_{2} = 2$
respectively
Also given that: $X = (\begin{matrix} 7 \\ 1 \end{matrix})$
To find: $A^{8} x = ?$
Step 2
To obtain: $A^{8} = ?$
So, $A = (\begin{array}{cc} 1 & 1 \\ - 1 & 1 \end{array}) \Rightarrow A^{2} = (\begin{array}{cc} 1 & 1 \\ - 1 & 1 \end{array}) (\begin{array}{cc} 1 & 1 \\ - 1 & 1 \end{array})$
$= (\begin{array}{cc} 0 & 2 \\ - 2 & 0 \end{array}) = 2 (\begin{array}{cc} 0 & 1 \\ - 1 & 0 \end{array})$
$A^{4} = A^{2} \times A^{2} = 2 (\begin{array}{cc} 0 & 1 \\ - 1 & 0 \end{array}) \times 2 (\begin{array}{cc} 0 & 1 \\ - 1 & 0 \end{array})$
$= 4 (\begin{array}{cc} 0 & 1 \\ - 1 & 0 \end{array}) \times (\begin{array}{cc} 0 & 1 \\ - 1 & 0 \end{array}) = 4 (\begin{array}{cc} - 1 & 0 \\ 0 & - 1 \end{array})$
Step 3
$A^{8} = A^{4} \times A^{4} = 4 (\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}) \times 4 (\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix})$
$= 16 (\begin{array}{cc} - 1 & 0 \\ 0 & - 1 \end{array}) \times (\begin{array}{cc} - 1 & 0 \\ 0 & - 1 \end{array}) = 16 = 16 (\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}) = 16 I$
Step 4
To obtain: $A^{8} X = ?$
$A^{8} X = 16 (\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}) (\begin{matrix} 7 \\ 1 \end{matrix}) = 16 (\begin{matrix} 7 \\ 1 \end{matrix}) = (\begin{matrix} 112 \\ 16 \end{matrix})$
Step 5
Result: $A^{8} X = (\begin{matrix} 112 \\ 16 \end{matrix})$

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