Solve the equation for $X$.
$X-3\left[\begin{array}{cc}
2 & -8 \\
-4 & 2
\end{array}\right]=\left[\begin{array}{cc}
4 & -6 \\
2 & -8
\end{array}\right]$
Answer (1)
Add 3 [ 2 − 8 − 4 2 ] to both sides of the equation.
Compute 3 [ 2 − 8 − 4 2 ] = [ 6 − 24 − 12 6 ] .
Compute X = [ 4 − 6 2 − 8 ] + [ 6 − 24 − 12 6 ] = [ 10 − 30 − 10 − 2 ] .
The solution is [ 10 − 30 − 10 − 2 ] .
Explanation
Understanding the Problem We are given the equation X − 3 [ 2 − 8 − 4 2 ] = [ 4 − 6 2 − 8 ] and we want to solve for the matrix X .
Isolating X To isolate X , we need to add 3 [ 2 − 8 − 4 2 ] to both sides of the equation. This gives us
X = [ 4 − 6 2 − 8 ] + 3 [ 2 − 8 − 4 2 ]
Scalar Multiplication First, let's compute the scalar multiplication:
3 [ 2 − 8 − 4 2 ] = [ 3 ( 2 ) 3 ( − 8 ) 3 ( − 4 ) 3 ( 2 ) ] = [ 6 − 24 − 12 6 ]
Matrix Addition Now, we add the two matrices:
X = [ 4 − 6 2 − 8 ] + [ 6 − 24 − 12 6 ] = [ 4 + 6 − 6 + ( − 24 ) 2 + ( − 12 ) − 8 + 6 ] = [ 10 − 30 − 10 − 2 ]
Final Answer Therefore, the solution for X is:
X = [ 10 − 30 − 10 − 2 ]
Examples
Matrix equations are used in computer graphics to perform transformations on objects. For example, to rotate, scale, or translate an object, a matrix is multiplied by the object's vertices. Solving matrix equations allows us to determine the transformations needed to achieve a desired effect. This is also applicable in robotics, where matrix equations are used to control the movement of robot arms and other mechanical systems.
