Write a matrix equation equivalent to the system of equations.
$\begin{array}{l}
2 x-4 y=-1 \\
2 x+4 y=4
\end{array}$
$\left[\begin{array}{rr}-1 & -4 \\
2 & 4\end{array}\right]\left[\begin{array}{l}x \\
y\end{array}\right]=\left[\begin{array}{l}2 \\
2\end{array}\right]$
$\left[\begin{array}{rr}2 & -4 \\
2 & 4\end{array}\right]\left[\begin{array}{l}x \\
y\end{array}\right]=\left[\begin{array}{r}-1 \\
4\end{array}\right]$
$\left[\begin{array}{rr}2 & -4 \\
2 & 4\end{array}\right]\left[\begin{array}{r}-1 \\
4\end{array}\right]=\left[\begin{array}{l}x \\
y\end{array}\right]$
$\left[\begin{array}{l}x \\
y\end{array}\right]\left[\begin{array}{rr}2 & -4 \\
2 & 4\end{array}\right]=\left[\begin{array}{r}-1 \\
4\end{array}\right]$
Answer (1)
Identify the coefficient matrix, variable matrix, and constant matrix from the given system of equations.
Form the matrix equation A X = B , where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
The coefficient matrix is [ 2 − 4 2 4 ] , the variable matrix is [ x y ] , and the constant matrix is [ − 1 4 ] .
The matrix equation equivalent to the system of equations is [ 2 − 4 2 4 ] [ x y ] = [ − 1 4 ] .
Explanation
Understanding the Problem We are given a system of two linear equations: 2 x − 4 y = − 1 and 2 x + 4 y = 4 . Our goal is to express this system in matrix form.
Matrix Form In matrix form, a system of linear equations can be written as A X = B , where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.
Coefficient Matrix The coefficient matrix A consists of the coefficients of the variables x and y in the given equations. Thus, A = [ 2 − 4 2 4 ] .
Variable Matrix The variable matrix X contains the variables x and y . Thus, X = [ x y ] .
Constant Matrix The constant matrix B contains the constants on the right-hand side of the equations. Thus, B = [ − 1 4 ] .
Final Matrix Equation Therefore, the matrix equation is [ 2 − 4 2 4 ] [ x y ] = [ − 1 4 ] .
The Answer Comparing this with the given options, we see that the correct matrix equation is [ 2 − 4 2 4 ] [ x y ] = [ − 1 4 ] .
Examples
Matrix equations are used in various fields such as computer graphics, physics, and engineering. For example, in computer graphics, transformations such as scaling, rotation, and translation of objects in 2D or 3D space can be represented using matrix equations. Solving these equations allows us to manipulate and display objects on the screen efficiently. In structural engineering, matrix equations are used to analyze the forces and stresses in complex structures, ensuring their stability and safety.
