Let $A = [\begin{array}{cc}a & 1 \\ 1 & b\end{array}]$, where $a$ and $b$ are the roots of the equation $x^2-4x+2 = 0$. If $A+A^{-1} = kI_2$, then the value of $k$ is ____

Question

Let the foot of the perpendicular from the point $ (1, 2, 4) $ on the line $ \frac{x+2}{4} = \frac{y-1}{2} = \frac{z+1}{3} $ be $ P $. Then the distance of $ P $ from the plane $ 3x + 4y + 12z + 23 = 0 $ is:

Answer 1

Step 1: Cayley-Hamilton route:
The characteristic equation of $A$ is $\lambda^2 - (a+b)\lambda + (ab - 1) = 0$, that is $\lambda^2 - 4\lambda + 1 = 0$.

Step 2: Apply to A:
$A^2 - 4A + I = 0$. Multiply by $A^{-1}$: $A - 4I + A^{-1} = 0$, so $A + A^{-1} = 4I$.

Step 3: Read off k:
$k = 4$.

Final Answer:
k equals 4, option (C). \[ \boxed{4} \]

Let \(A = [\begin{array}{cc}a & 1 \\ 1 & b\end{array}]\), where \(a\) and \(b\) are the roots of the equation \(x^2-4x+2 = 0\). If \(A+A^{-1} = kI_2\), then the value of \(k\) is ____

Show Hint

The Correct Option is C

Solution and Explanation

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