Question:
If $A = \begin{bmatrix}a&b\\ b&a\end{bmatrix}$ and $A^{2}\begin{bmatrix}\alpha&\beta \\ \beta&\alpha\end{bmatrix}$, then
If $A = \begin{bmatrix}a&b\\ b&a\end{bmatrix}$ and $A^{2}\begin{bmatrix}\alpha&\beta \\ \beta&\alpha\end{bmatrix}$, then
Updated On: Jul 28, 2022
- $\alpha = 2ab , \beta = a^2 + b^2$
- $\alpha = a^2 + b^2 , \beta = ab $
- $\alpha = a^2 + b^2 , \beta = 2ab$
- $\alpha = a^2 + b^2 , \beta = a^2 - b^2$
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The Correct Option is C
Solution and Explanation
$ A^{2}\begin{bmatrix}\alpha&\beta \\ \beta&\alpha\end{bmatrix} $
$ \Rightarrow\begin{bmatrix}\alpha&\beta\\ \beta&\alpha\end{bmatrix} = \begin{bmatrix}a&b\\ b&a\end{bmatrix}\begin{bmatrix}a&b\\ b&a\end{bmatrix} $
$ \Rightarrow \begin{bmatrix}\alpha&\beta\\ \beta&\alpha\end{bmatrix} = \begin{bmatrix}a^{2} +b^{2}&2ab\\ 2ab&a^{2}+b^{2}\end{bmatrix} $
$ \Rightarrow \alpha = a^{2}+b^{2}$ and $ 2ab = \beta$
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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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- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.