What is the solution of the matrix equation?
[tex]
\left[\begin{array}{l}
x \\
y
\end{array}\right]=\left[\begin{array}{ll}
-3 & 5 \\
-1 & 2
\end{array}\right]\left[\begin{array}{l}
0 \\
2
\end{array}\right]
[/tex]
Answer (1)
Perform the matrix multiplication: [ − 3 5 − 1 2 ] [ 0 2 ] = [ ( − 3 ) ( 0 ) + ( 5 ) ( 2 ) ( − 1 ) ( 0 ) + ( 2 ) ( 2 ) ] .
Simplify the result: [ 0 + 10 0 + 4 ] = [ 10 4 ] .
Identify the values of x and y : x = 10 and y = 4 .
State the solution: ( 10 , 4 ) .
Explanation
Understanding the Problem We are given a matrix equation and asked to find the solution. The equation is: [ x y ] = [ − 3 5 − 1 2 ] [ 0 2 ] To solve this, we need to perform the matrix multiplication on the right side of the equation.
Performing Matrix Multiplication Let's perform the matrix multiplication: [ x y ] = [ ( − 3 ) ( 0 ) + ( 5 ) ( 2 ) ( − 1 ) ( 0 ) + ( 2 ) ( 2 ) ] This simplifies to: [ x y ] = [ 0 + 10 0 + 4 ] [ x y ] = [ 10 4 ]
Finding the Solution From the resulting matrix, we can see that: x = 10 and y = 4 .
Therefore, the solution to the matrix equation is the vector [ 10 4 ] .
Final Answer Thus, the solution to the matrix equation is ( 10 , 4 ) .
Examples
Matrix equations are used in various fields such as computer graphics, physics, and engineering. For example, in computer graphics, transformations like scaling, rotation, and translation of objects in 2D or 3D space can be represented using matrices. Solving a matrix equation can help determine the final position of an object after a series of transformations. This is crucial for rendering images and animations in video games and movies.
