Question:
If $\begin{bmatrix}3&1\\ 4&1\end{bmatrix} X = \begin{bmatrix}5&-1\\ 2&3\end{bmatrix} $ then X is equal to:
If $\begin{bmatrix}3&1\\ 4&1\end{bmatrix} X = \begin{bmatrix}5&-1\\ 2&3\end{bmatrix} $ then X is equal to:
Updated On: Jul 6, 2022
- $\begin{bmatrix}-3&4\\ 14&-13\end{bmatrix}$
- $ \begin{bmatrix}3&-4\\ -14&13\end{bmatrix}$
- $\begin{bmatrix}3&4\\ 14&13\end{bmatrix}$
- $\begin{bmatrix}-3&4\\ -14&13\end{bmatrix}$
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The Correct Option is A
Solution and Explanation
Given : $\begin{bmatrix}3&1\\ 4&1\end{bmatrix} X = \begin{bmatrix}5&-1\\ 2&3\end{bmatrix} $
Thus $X = \begin{bmatrix}3&1\\ 4&1\end{bmatrix}^{-1} \begin{bmatrix}5&-1\\ 2&3\end{bmatrix}$
Let $ A = \begin{bmatrix}3&1\\ 4&1\end{bmatrix} $
$a_{11}$ = co-factor of $a_{11} = 1$
$a_{12}$ = co-factor of $a_{12} = (-1)^{1+2}. 4 = - 4$
$a_{21}$ = co-factor of $a_{21} = (-1)^{2+1} . 1 = - 1$
$a_{22}$ = co-factor of $a_{22} = 3 $
$| A | = 3 - 4 = -1 $
So $ A^{-1} = \frac{\begin{bmatrix}1&-4\\ -1&3\end{bmatrix}}{-1} = \begin{bmatrix}-1&1\\ 4&-3\end{bmatrix}$
Thus $ X = \begin{bmatrix}-1&1\\ 4&-13\end{bmatrix}\begin{bmatrix}5&-1\\ 2&3\end{bmatrix} $
$ X = \begin{bmatrix}-3&4\\ 14&-13\end{bmatrix} $
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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.