Solve the following matrix equation. If an equation cannot be solved, explain why.
4 5 3 -1
3 4 x= 4 3
To solve the matrix equation, we need to find the matrix x.
Let's set up the equation in matrix form:
|4 5| |x1 x2| |3 -1|
|3 4| x |x3 x4| = |4 3|
Multiplying the matrices, we get:
4x1 + 5x3 = 3
3x1 + 4x3 = 4
Solving this system of equations, we can use various methods (substitution, elimination, matrix inverse). Here, we can use substitution.
From the first equation, solve for x3:
4x3 = (3 - 4x1)
x3 = (3/4) - x1
Substitute this expression for x3 into the second equation:
3x1 + 4((3/4) - x1) = 4
3x1 + 3 - 4x1 = 4
-x1 = 1
x1 = -1
Now substitute x1 = -1 into the first equation:
4(-1) + 5x3 = 3
-4 + 5x3 = 3
5x3 = 7
x3 = 7/5
Thus, we have found values for x1 and x3:
x1 = -1
x3 = 7/5
Now we can substitute these values back into the second equation to find x2 and x4.
3(-1) + 4x2 = 4
-3 + 4x2 = 4
4x2 = 7
x2 = 7/4
4((-1) + 5x4) = 3
-4 + 20x4 = 3
20x4 = 7
x4 = 7/20
Therefore, the solution for matrix x is:
|x1 x2| |-1 7/4|
|x3 x4| = |7/5 7/20|
So, the matrix equation is solvable and the solution is:
|-1 7/4|
|7/5 7/20|