If \(A=\begin{bmatrix}1 & 0\\2 & 1\end{bmatrix}\) then value of \(\begin{bmatrix}1 & 0\\2 & 1\end{bmatrix}^{2007}\)
\(\begin{bmatrix}1 & 0\\4014 & 1\end{bmatrix}\)
Write \(A=I+N\), where \(I=\begin{bmatrix}1&0\\0&1\end{bmatrix}\) and \(N=\begin{bmatrix}0&0\\2&0\end{bmatrix}\).
Compute \(N^2=\begin{bmatrix}0&0\\2&0\end{bmatrix}\begin{bmatrix}0&0\\2&0\end{bmatrix}=\begin{bmatrix}0&0\\0&0\end{bmatrix}\), so N is nilpotent with \(N^2=0\).
Since I and N commute, by the binomial theorem \(A^n=(I+N)^n=I+nN+\binom{n}{2}N^2+\cdots=I+nN\) (all higher terms vanish because \(N^2=0\)).
So \(A^n=I+nN=\begin{bmatrix}1&0\\2n&1\end{bmatrix}\).
For \(n=2007\): \(A^{2007}=\begin{bmatrix}1&0\\2\times2007&1\end{bmatrix}=\begin{bmatrix}1&0\\4014&1\end{bmatrix}\).
Similar Questions
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 \(2\times2\))
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?
