Solve the following equation.
$2\left(\begin{array}{lll}
1 & -1 & 3 \\
2 & -7 & 5
\end{array}\right)+X=3\left(\begin{array}{rrr}
1 & 2 & -4 \\
3 & -5 & 1
\end{array}\right)$
8. Solve : $\left(\begin{array}{ll}1 & 2 \\
3 & 4\end{array}\right)-3 X=2\left(\begin{array}{rr}-4 & 7 \\
3 & 8\end{array}\right)$
9. Solve: $\left(\begin{array}{ll}2 & 3 \\
4 & 5\end{array}\right)+ X =\left(\begin{array}{rr}4 & -1 \\
3 & 2\end{array}\right)$
10. Solve the following equations.
(a) $2 X+\left(\begin{array}{rr}3 & 4 \\
-6 & 1\end{array}\right)=\left(\begin{array}{ll}7 & 6 \\
2 & 1\end{array}\right)$
(b) $\left(\begin{array}{rr}-2 & -1 \\
5 & 0\end{array}\right)-3 X=\left(\begin{array}{rr}4 & -8 \\
2 & 6\end{array}\right)$
Answer (1)
Equation 7: Isolate X and perform matrix operations to find X = ( 1 5 8 − 1 − 18 − 7 ) .
Equation 8: Isolate X and perform matrix operations to find X = ( 3 − 1 − 4 − 4 ) .
Equation 9: Isolate X and perform matrix operations to find X = ( 2 − 1 − 4 − 3 ) .
Equation 10(a): Isolate X and perform matrix operations to find X = ( 2 4 1 0 ) .
Equation 10(b): Isolate X and perform matrix operations to find X = ( − 2 1 3 7 − 2 ) .
Explanation
Problem Analysis We are given several matrix equations to solve for the unknown matrix X . We will isolate X in each equation and then perform the necessary matrix operations (scalar multiplication and matrix addition/subtraction) to find the solution.
Solving Equation 7 For equation 7, we have: 2 ( 1 2 − 1 − 7 3 5 ) + X = 3 ( 1 3 2 − 5 − 4 1 ) Isolating X , we get: X = 3 ( 1 3 2 − 5 − 4 1 ) − 2 ( 1 2 − 1 − 7 3 5 ) Performing scalar multiplication: X = ( 3 9 6 − 15 − 12 3 ) − ( 2 4 − 2 − 14 6 10 ) Performing matrix subtraction: X = ( 3 − 2 9 − 4 6 − ( − 2 ) − 15 − ( − 14 ) − 12 − 6 3 − 10 ) = ( 1 5 8 − 1 − 18 − 7 )
Solving Equation 8 For equation 8, we have: ( 1 3 2 4 ) − 3 X = 2 ( − 4 3 7 8 ) Isolating 3 X , we get: 3 X = ( 1 3 2 4 ) − 2 ( − 4 3 7 8 ) Performing scalar multiplication: 3 X = ( 1 3 2 4 ) − ( − 8 6 14 16 ) Performing matrix subtraction: 3 X = ( 1 − ( − 8 ) 3 − 6 2 − 14 4 − 16 ) = ( 9 − 3 − 12 − 12 ) Multiplying by 3 1 :
X = 3 1 ( 9 − 3 − 12 − 12 ) = ( 3 − 1 − 4 − 4 )
Solving Equation 9 For equation 9, we have: ( 2 4 3 5 ) + X = ( 4 3 − 1 2 ) Isolating X , we get: X = ( 4 3 − 1 2 ) − ( 2 4 3 5 ) Performing matrix subtraction: X = ( 4 − 2 3 − 4 − 1 − 3 2 − 5 ) = ( 2 − 1 − 4 − 3 )
Solving Equation 10(a) For equation 10(a), we have: 2 X + ( 3 − 6 4 1 ) = ( 7 2 6 1 ) Isolating 2 X , we get: 2 X = ( 7 2 6 1 ) − ( 3 − 6 4 1 ) Performing matrix subtraction: 2 X = ( 7 − 3 2 − ( − 6 ) 6 − 4 1 − 1 ) = ( 4 8 2 0 ) Multiplying by 2 1 :
X = 2 1 ( 4 8 2 0 ) = ( 2 4 1 0 )
Solving Equation 10(b) For equation 10(b), we have: ( − 2 5 − 1 0 ) − 3 X = ( 4 2 − 8 6 ) Isolating 3 X , we get: 3 X = ( − 2 5 − 1 0 ) − ( 4 2 − 8 6 ) Performing matrix subtraction: 3 X = ( − 2 − 4 5 − 2 − 1 − ( − 8 ) 0 − 6 ) = ( − 6 3 7 − 6 ) Multiplying by 3 1 :
X = 3 1 ( − 6 3 7 − 6 ) = ( − 2 1 3 7 − 2 )
Examples
Matrix equations are used in various fields such as computer graphics, physics, and engineering. For example, in computer graphics, matrices are used to represent transformations such as rotation, scaling, and translation of objects in 3D space. Solving matrix equations allows us to determine the necessary transformations to achieve a desired effect. In structural engineering, matrices are used to analyze the forces and stresses within a structure. Solving matrix equations helps engineers to ensure the stability and safety of buildings and bridges.
