Question:

If \(\begin{bmatrix}e^{x}&e^{y}\\ e^{y}&e^{x}\end{bmatrix} = \begin{bmatrix}1&1\\ 1&1\end{bmatrix}\), then the values of \(x\) and \(y\) are respectively:

Updated On: Jun 7, 2024
  • -1,-1
  • 0, 0

  • 0, 1
  • 1, 0
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The Correct Option is B

Solution and Explanation

Given, \(\begin{bmatrix}e^{x} & e^{y} \\ e^{y} & e^{x}\end{bmatrix}=\begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}\)
\(\Rightarrow \begin{bmatrix}e^{x} & e^{y} \\ e^{y} & e^{x}\end{bmatrix}=\begin{bmatrix}e^{0} & e^{0} \\ e^{0} & e^{0}\end{bmatrix} (\because e^{0}=1)\)
On equating the corresponding elements,
\(e^{x}=e^{0}\) and \(e^{y}=e^{0}\)
\(\Rightarrow x=0\) and \(y=0\)

So, the correct option is (B): 0, 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:

  1. 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.
  2. Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
  3. 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. 
  4. 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. 
  5. Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.