What is the solution of the matrix equation $\left[\begin{array}{ll}1 & 3 \\ 1 & 4\end{array}\right] X=\left[\begin{array}{l}6 \\ 8\end{array}\right]$?
A. $X=\left[\begin{array}{c}
0 \
-2
\end{array}\right]$
B. $X=\left[\begin{array}{c}
-2 \
0
\end{array}\right]$
C. $X=\left[\begin{array}{l}
0 \
2
\end{array}\right]$
D. $X=\left[\begin{array}{l}
2 \
0
\end{array}\right]$
Answer (2)
Calculate the determinant of the matrix [ 1 3 1 4 ] which is equal to 1.
Find the inverse of the matrix [ 1 3 1 4 ] which is [ 4 − 3 − 1 1 ] .
Multiply the inverse by the matrix [ 6 8 ] to find the solution.
The solution of the matrix equation is X = [ 0 2 ] , so the answer is [ 0 2 ] .
Explanation
Understanding the Problem We are given the matrix equation [ 1 3 1 4 ] X = [ 6 8 ] and we need to find the matrix X .
Setting up the Solution Let A = [ 1 3 1 4 ] and B = [ 6 8 ] . We want to solve A X = B for X . To do this, we can find the inverse of A , denoted by A − 1 , and multiply both sides of the equation by A − 1 to get X = A − 1 B .
Finding the Determinant First, we find the determinant of A :
det ( A ) = ( 1 ) ( 4 ) − ( 3 ) ( 1 ) = 4 − 3 = 1.
Finding the Inverse Next, we find the inverse of A :
A − 1 = det ( A ) 1 [ 4 − 3 − 1 1 ] = [ 4 − 3 − 1 1 ] .
Calculating X Now, we can find X by multiplying A − 1 by B :
X = A − 1 B = [ 4 − 3 − 1 1 ] [ 6 8 ] = [ ( 4 ) ( 6 ) + ( − 3 ) ( 8 ) ( − 1 ) ( 6 ) + ( 1 ) ( 8 ) ] = [ 24 − 24 − 6 + 8 ] = [ 0 2 ] .
Final Answer Therefore, the solution to the matrix equation is X = [ 0 2 ] .
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 3D space can be represented using matrices. Solving matrix equations allows us to determine the parameters of these transformations or to find the coordinates of transformed objects. In structural engineering, matrix equations are used to analyze the forces and displacements in complex structures.
The solution to the matrix equation [ 1 1 3 4 ] X = [ 6 8 ] is X = [ 0 2 ] . The correct answer is option C. The matrix inversion and multiplication confirm this result.
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