Question:
if $\begin{bmatrix}x\\ y\\ z\end{bmatrix} = \frac{1}{40} \begin{bmatrix}5&10&-5\\ -5&-2&13\\ 10&-4&6\end{bmatrix}\begin{bmatrix}5\\ 0\\ 5\end{bmatrix}$, then the value of $x + y + z$ is
if $\begin{bmatrix}x\\ y\\ z\end{bmatrix} = \frac{1}{40} \begin{bmatrix}5&10&-5\\ -5&-2&13\\ 10&-4&6\end{bmatrix}\begin{bmatrix}5\\ 0\\ 5\end{bmatrix}$, then the value of $x + y + z$ is
Updated On: Jul 6, 2022
- $3$
- $0$
- $2$
- $1$
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The Correct Option is A
Solution and Explanation
Given$\begin{bmatrix}x\\ y\\ z\end{bmatrix} = \frac{1}{40} \begin{bmatrix}5&10&-5\\ -5&-2&13\\ 10&-4&6\end{bmatrix}\begin{bmatrix}5\\ 0\\ 5\end{bmatrix}$
$= \frac{1}{40} \begin{bmatrix}25&+0&-25\\ -25&+0&+65\\ 50&+0&+30\end{bmatrix}= \frac{1}{40}\begin{bmatrix}0\\ 40\\ 80\end{bmatrix}= \begin{bmatrix}0\\ 1\\ 2\end{bmatrix}$
$\Rightarrow x = 0, y = 1, z = 2$
$\therefore x + y + z = 0 + 1 + 2 = 3$
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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.