Question:
If $A = \begin{pmatrix}1&5\\ 0&2\end{pmatrix}$ , then
If $A = \begin{pmatrix}1&5\\ 0&2\end{pmatrix}$ , then
Updated On: Jun 6, 2024
- $A^2 - 2A + 2I = 0$
- $A^2 - 3A + 2I = 0$
- $A^2 - 5A + 2I = 0$
- $2A^2 - A + I = 0$
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The Correct Option is B
Solution and Explanation
We have,
$A=\begin{bmatrix}1 & 5 \\ 0 & 2\end{bmatrix}$
$ \therefore A^{2}=A \cdot A=\begin{bmatrix}1 & 5 \\ 0 & 2\end{bmatrix}\begin{bmatrix}1 & 5 \\ 0 & 2\end{bmatrix}=\begin{bmatrix}1 & 15 \\ 0 & 4\end{bmatrix}$
$\therefore A^{2}-3 A+2 I=\begin{bmatrix}1 & 15 \\ 0 & 4\end{bmatrix}-3\begin{bmatrix}1 & 5 \\ 0 & 2\end{bmatrix}$
$+2\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}=\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}$
$\Rightarrow A^{2}-3 A+2 I=0$
$A=\begin{bmatrix}1 & 5 \\ 0 & 2\end{bmatrix}$
$ \therefore A^{2}=A \cdot A=\begin{bmatrix}1 & 5 \\ 0 & 2\end{bmatrix}\begin{bmatrix}1 & 5 \\ 0 & 2\end{bmatrix}=\begin{bmatrix}1 & 15 \\ 0 & 4\end{bmatrix}$
$\therefore A^{2}-3 A+2 I=\begin{bmatrix}1 & 15 \\ 0 & 4\end{bmatrix}-3\begin{bmatrix}1 & 5 \\ 0 & 2\end{bmatrix}$
$+2\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}=\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}$
$\Rightarrow A^{2}-3 A+2 I=0$
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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.