If the matrix \(A=\begin{bmatrix}1&0&0\\0&2&0\\3&0&-1\end{bmatrix}\) satisfies the equation
\(A^{20}+\alpha A^{19}+\beta A=\begin{bmatrix}1&0&0\\0&4&0\\0&0&1\end{bmatrix}\)
for some real numbers \(\alpha\) and \(\beta\), then \(\beta-\alpha\) is equal to ___________.
\(A^{20}+\alpha A^{19}+\beta A=\begin{bmatrix}1&0&0\\0&4&0\\0&0&1\end{bmatrix}\)
for some real numbers \(\alpha\) and \(\beta\), then \(\beta-\alpha\) is equal to ___________.
Show Hint
Correct Answer: 4
Solution and Explanation
Block 1,3: \(I+\alpha B+\beta B=I\Rightarrow(\alpha+\beta)B=0\Rightarrow\alpha+\beta=0\).
Entry (2,2): \(2^{20}+\alpha2^{19}+2\beta=4\). With \(\beta=-\alpha\): \(\alpha(2^{19}-2)=4-2^{20}=-2(2^{19}-2)\Rightarrow\alpha=-2\), \(\beta=2\).
\(\beta-\alpha=2-(-2)=4\).
Answer: 4
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