Question:
If p, q, r are 3 real numbers satisfying the matrix equation , $[ p \ q \ r ] \begin{bmatrix}3&4&1\\ 3&2&3\\ 2&0&2\end{bmatrix} = \begin{bmatrix}3&0&1\end{bmatrix}$ then 2p + q - r equals :
If p, q, r are 3 real numbers satisfying the matrix equation , $[ p \ q \ r ] \begin{bmatrix}3&4&1\\ 3&2&3\\ 2&0&2\end{bmatrix} = \begin{bmatrix}3&0&1\end{bmatrix}$ then 2p + q - r equals :
Updated On: Jan 30, 2024
- -3
- -1
- 4
- 2
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The Correct Option is A
Solution and Explanation
Given
$\begin{bmatrix}p&q&r\end{bmatrix} \begin{bmatrix}3&4&1\\ 3&2&3\\ 2&0&2\end{bmatrix} = \begin{bmatrix}3&0&1\end{bmatrix} $
$ \Rightarrow \begin{bmatrix}3p + 3q + 2r&4p + 2q&p + 3q + 2r\end{bmatrix} = \begin{bmatrix}3&0&1\end{bmatrix}$
$ \Rightarrow \ 3p + 3q + 2r = 3 $ ...(i)
$4p + 2q = 0 \ \Rightarrow \ q = - 2p$ ...(ii)
$p + 3q + 2r = 1$ ...(iii)
On solving (i), (ii) and (iii), we get
p = 1, q = - 2, r = 3
$\therefore$ 2p + q - r = 2(1) + (- 2) - (3) = - 3.
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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.