+0

0
112
1
+20

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
+25656
+3

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?

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

Sep 28, 2020