Define \(A=\begin{bmatrix}1 & 1\\3 & 0\end{bmatrix}\). Find a vertical vector V such that \((A^8+A^6+A^4+A^2+I)V=\begin{bmatrix}0\\11\end{bmatrix}\) (Where I is the identity matrix of order \(2\times2\))
\(\begin{bmatrix}0\\1/10\end{bmatrix}\)
The characteristic equation of \(A=\begin{bmatrix}1&1\\3&0\end{bmatrix}\) is \(\lambda^2-\lambda-3=0\), so by the Cayley-Hamilton theorem \(A^2=A+3I\).
Writing every power as \(A^n=a_nA+b_nI\) with \(a_{n+1}=a_n+b_n,\ b_{n+1}=3a_n\) and \(a_1=1,b_1=0\), computing successively gives (A²,A⁴,A⁶,A⁸) with coefficients \((a,b)\) equal to \((1,3),(7,12),(40,57),(217,291)\) respectively.
Summing \(A^8+A^6+A^4+A^2+I\): coefficient of A is \(217+40+7+1=265\); coefficient of I is \(291+57+12+3+1=364\). So \(S=265A+364I=\begin{bmatrix}629&265\\795&364\end{bmatrix}\).
Solving \(S\,V=\begin{bmatrix}0\\11\end{bmatrix}\) exactly by Cramer's rule (with \(\det S=18281\)) gives \(v_1=-\dfrac{2915}{18281}\), \(v_2=\dfrac{6919}{18281}\), which does not correspond exactly to any of the printed options (all of which require \(v_1=0\)).
Since \(S\) is not lower-triangular (its (1,2) entry, 265, is non-zero), no vector with a zero first entry can satisfy this system exactly, indicating an inconsistency between the given matrix/answer options in the source paper. Based on the closest-magnitude option to the exact second component, option (A) is selected, but this answer could not be fully verified against the printed choices.
Similar Questions
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Important Questions from Operations on Matrices
If \(A=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]\), then what is the value of det(I + AA'), where I is the 3 × 3 identity matrix?
If \(A=\left[\begin{array}{lll} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{array}\right]\), then which of the following statements are correct?
1. An will always be singular for any positive integer n.
2. An will always be a diagonal matrix for any positive integer n.
3. An will always be a symmetric matrix for any positive integer n.
Select the correct answer using the code given below:
If \(A=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\), then what is 23A3 - 19A2 - 4A equal to ?
If \(A_k=\left[\begin{array}{cc} k-1 & k \\ k-2 & k+1 \end{array}\right] \), then what is det(A1) + det(A2) + det(A3) + ... + det(A100) equal to ?
Consider the following in respect of the matrix \({\rm{A}} = \left( {\begin{array}{*{20}{c}} { - 1}&1\\ 1&{ - 1} \end{array}} \right):\)
1. A 2= -A
2. A 3= 4A
Which of the above is/are correct?
