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Let A be a matrix and x be a non-zero vector such that \(\mathbf{A} \mathbf{x} = 2 \mathbf{x}\)
Then we have that \(\mathbf{A}^7\mathbf{x} = a \mathbf{x}\)
some value of a. What is a equal to?

 Sep 25, 2020
 #1
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Let A be a matrix and x be a non-zero vector such that \(\mathbf{A} \mathbf{x} = 2 \mathbf{x}\)
Then we have that \(\mathbf{A}^7\mathbf{x} = a \mathbf{x}\)
some value of a. What is a equal to?

 

\(\begin{array}{|rcll|} \hline \mathbf{A}^7\mathbf{x} &=& \mathbf{A}^6A\mathbf{x} \quad | \quad \mathbf{A} \mathbf{x} = 2 \mathbf{x} \\ &=& A^6 2x \\ &=& 2A^6 x \\ &=& 2A^5A x \quad | \quad \mathbf{A} \mathbf{x} = 2 \mathbf{x} \\ &=& 2A^52x \\ &=& 4A^5x \\ &=& 4A^4Ax \quad | \quad \mathbf{A} \mathbf{x} = 2 \mathbf{x} \\ &=& 4A^42x \\ &=& 8A^4x \\ &=& 8A^3Ax \quad | \quad \mathbf{A} \mathbf{x} = 2 \mathbf{x} \\ &=& 8A^32x \\ &=& 16A^3x \\ &=& 16A^2Ax \quad | \quad \mathbf{A} \mathbf{x} = 2 \mathbf{x} \\ &=& 32A^2x \\ &=& 32AAx \quad | \quad \mathbf{A} \mathbf{x} = 2 \mathbf{x} \\ &=& 32A2x \\ &=& 64Ax\quad | \quad \mathbf{A} \mathbf{x} = 2 \mathbf{x} \\ &=& 64A2x \\ \mathbf{A}^7\mathbf{x}&=& \mathbf{128x} \\ \hline \end{array}\)

 

laugh

 Sep 28, 2020

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