Question:
If $ A=\left[ \begin{matrix} 2 & -1 \\ 1 & 2 \\ \end{matrix} \right], $ then $ {{A}^{2}}+2A-3I= $
If $ A=\left[ \begin{matrix} 2 & -1 \\ 1 & 2 \\ \end{matrix} \right], $ then $ {{A}^{2}}+2A-3I= $
Updated On: Jun 23, 2024
- $ \left[ \begin{matrix} 4 & -6 \\ 6 & 4 \\ \end{matrix} \right] $
- $ 0 $
- $ \left[ \begin{matrix} -6 & 2 \\ -2 & 6 \\ \end{matrix} \right] $
- $ 5I $
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The Correct Option is A
Solution and Explanation
Given, $ A=\left[ \begin{matrix} 2 & -1 \\ 1 & 2 \\ \end{matrix} \right] $ Now, $ {{A}^{2}}=\left[ \begin{matrix} 2 & -1 \\ 1 & 2 \\ \end{matrix} \right]\left[ \begin{matrix} 2 & -1 \\ 1 & 2 \\ \end{matrix} \right]=\left[ \begin{matrix} 4-1 & -2-2 \\ 2+2 & -1+4 \\ \end{matrix} \right] $
$ \Rightarrow $ $ {{A}^{2}}=\left[ \begin{matrix} 3 & -4 \\ 4 & 3 \\ \end{matrix} \right] $
Now, $ {{A}^{2}}+2A-3l=\left[ \begin{matrix} 3 & -4 \\ 4 & 3 \\ \end{matrix} \right]+2\left[ \begin{matrix} 2 & -1 \\ 1 & 2 \\ \end{matrix} \right]-3\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right] $
$=\left[ \begin{matrix} 3+4-3-4-2-0 \\ 4+2-0\,\,3+4-3 \\ \end{matrix} \right]\,=\,\left[ \begin{matrix} 4 & -6 \\ 6 & 4 \\ \end{matrix} \right] $
$ \Rightarrow $ $ {{A}^{2}}=\left[ \begin{matrix} 3 & -4 \\ 4 & 3 \\ \end{matrix} \right] $
Now, $ {{A}^{2}}+2A-3l=\left[ \begin{matrix} 3 & -4 \\ 4 & 3 \\ \end{matrix} \right]+2\left[ \begin{matrix} 2 & -1 \\ 1 & 2 \\ \end{matrix} \right]-3\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right] $
$=\left[ \begin{matrix} 3+4-3-4-2-0 \\ 4+2-0\,\,3+4-3 \\ \end{matrix} \right]\,=\,\left[ \begin{matrix} 4 & -6 \\ 6 & 4 \\ \end{matrix} \right] $
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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.
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