Question:
If $n$ is a non-negative integer and $A = \begin{bmatrix}1&0\\ 1&1\end{bmatrix} $ , then $A^n = $
If $n$ is a non-negative integer and $A = \begin{bmatrix}1&0\\ 1&1\end{bmatrix} $ , then $A^n = $
Updated On: May 11, 2024
- $\begin{bmatrix}1&0\\ n - 1&1\end{bmatrix} $
- $\begin{bmatrix}1&0\\1&1\end{bmatrix} $
- $\begin{bmatrix}1&0\\n &1\end{bmatrix} $
- $\begin{bmatrix}1&n\\0&1\end{bmatrix} $
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The Correct Option is C
Solution and Explanation
$A = \begin{bmatrix}1&0\\ 1&1\end{bmatrix}$
$ A^{2} = \begin{bmatrix}1&0\\ 1&1\end{bmatrix}\begin{bmatrix}1&0\\ 1&1\end{bmatrix} =\begin{bmatrix}1&0\\ 2&1\end{bmatrix} $
Similarly, $A^{3} =\begin{bmatrix}1&0\\ 3&1\end{bmatrix} $
$A^{4} = \begin{bmatrix}1&0\\ 4&1\end{bmatrix}$ and so on
Hence , $A^{n}= \begin{bmatrix}1&0\\ n&1\end{bmatrix} $
$ A^{2} = \begin{bmatrix}1&0\\ 1&1\end{bmatrix}\begin{bmatrix}1&0\\ 1&1\end{bmatrix} =\begin{bmatrix}1&0\\ 2&1\end{bmatrix} $
Similarly, $A^{3} =\begin{bmatrix}1&0\\ 3&1\end{bmatrix} $
$A^{4} = \begin{bmatrix}1&0\\ 4&1\end{bmatrix}$ and so on
Hence , $A^{n}= \begin{bmatrix}1&0\\ n&1\end{bmatrix} $
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Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.