Question:
If $X$ and $Y$ are $ 2\times 2 $ matrices such that $ 2X+3Y=O $ and $ X+2Y=I, $ where $ O $ and $ I $ denote the $ 2\times 2 $ zero matrix and the $ 2\times 2 $ identity matrix, then $X$ is equal to
If $X$ and $Y$ are $ 2\times 2 $ matrices such that $ 2X+3Y=O $ and $ X+2Y=I, $ where $ O $ and $ I $ denote the $ 2\times 2 $ zero matrix and the $ 2\times 2 $ identity matrix, then $X$ is equal to
Updated On: Jun 23, 2024
- $ \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right] $
- $ \left[ \begin{matrix} 2 & 0 \\ 0 & 2 \\ \end{matrix} \right] $
- $ \left[ \begin{matrix} -3 & 0 \\ 0 & -3 \\ \end{matrix} \right] $
- $ \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \\ \end{matrix} \right] $
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The Correct Option is C
Solution and Explanation
Given, $ 2X+3Y=O $ .....(i) and $ X+2Y=I $ ..(ii)
Where $ O=\left[ \begin{matrix} 0 & 0 \\ 0 & 0 \\ \end{matrix} \right] $ and $ I=\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right] $
On solving Eqs. (i) and (ii), we get
$ X=-3I=\left[ \begin{matrix} -3 & 0 \\ 0 & -3 \\ \end{matrix} \right] $
Where $ O=\left[ \begin{matrix} 0 & 0 \\ 0 & 0 \\ \end{matrix} \right] $ and $ I=\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right] $
On solving Eqs. (i) and (ii), we get
$ X=-3I=\left[ \begin{matrix} -3 & 0 \\ 0 & -3 \\ \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.
- 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.