Question:
If $A = \begin{pmatrix}1&0\\ 1&1\end{pmatrix}$ , then $A^n + nI$ is equal to
If $A = \begin{pmatrix}1&0\\ 1&1\end{pmatrix}$ , then $A^n + nI$ is equal to
Updated On: Jun 7, 2024
- I
- nA
- I + nA
- I - nA
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The Correct Option is C
Solution and Explanation
We have
$A=\begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}$
$\therefore A^{2}=A \cdot A=\begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}\begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}=\begin{bmatrix}1 & 0 \\ 2 & 1\end{bmatrix}$
$A^{3}=A^{2} \cdot A=\begin{bmatrix}1 & 0 \\ 2 & 1\end{bmatrix}\begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}=\begin{bmatrix}1 & 0 \\ 3 & 1\end{bmatrix}$
$\therefore A^{n}=\begin{bmatrix}1 & 0 \\ n & 1\end{bmatrix}$
Now,
$A^{n}+n I=\begin{bmatrix}1 & 0 \\ n & 1\end{bmatrix}+n\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$
$=\begin{bmatrix}1 & 0 \\ n & 1\end{bmatrix}+\begin{bmatrix}n & 0 \\ 0 & n\end{bmatrix}$
$=\begin{bmatrix}1+n & 0 \\ n & 1+n\end{bmatrix}$
Again, $I+n A=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}+n\begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}$
$=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}+\begin{bmatrix}n & 0 \\ n & n\end{bmatrix}$
$=\begin{bmatrix}1+n & 0 \\ n & 1+n\end{bmatrix}$
$\therefore A^{n}+n I=I+ n A$
$A=\begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}$
$\therefore A^{2}=A \cdot A=\begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}\begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}=\begin{bmatrix}1 & 0 \\ 2 & 1\end{bmatrix}$
$A^{3}=A^{2} \cdot A=\begin{bmatrix}1 & 0 \\ 2 & 1\end{bmatrix}\begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}=\begin{bmatrix}1 & 0 \\ 3 & 1\end{bmatrix}$
$\therefore A^{n}=\begin{bmatrix}1 & 0 \\ n & 1\end{bmatrix}$
Now,
$A^{n}+n I=\begin{bmatrix}1 & 0 \\ n & 1\end{bmatrix}+n\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$
$=\begin{bmatrix}1 & 0 \\ n & 1\end{bmatrix}+\begin{bmatrix}n & 0 \\ 0 & n\end{bmatrix}$
$=\begin{bmatrix}1+n & 0 \\ n & 1+n\end{bmatrix}$
Again, $I+n A=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}+n\begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}$
$=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}+\begin{bmatrix}n & 0 \\ n & n\end{bmatrix}$
$=\begin{bmatrix}1+n & 0 \\ n & 1+n\end{bmatrix}$
$\therefore A^{n}+n I=I+ n A$
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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.