What is the solution of the matrix equation?
[9 4];[2 1]X=[-9 -6];[-1 -8]
The matrix equation [9 4; 2 1]X = [-9 -6; -1 -8] can be rewritten as:
9x + 4y = -9
2x + y = -6
We can solve this system of equations using various methods, such as substitution or elimination. Let's use the elimination method to solve it.
First, let's multiply the second equation by 4 to make the coefficients of y in both equations equal:
8x + 4y = -24
Now, subtract the first equation from this new equation:
(8x + 4y) - (9x + 4y) = -24 - (-9)
Simplifying:
- x = -15
Now, substitute this value of x back into one of the original equations (let's use the first equation):
9(-15) + 4y = -9
Simplifying:
-135 + 4y = -9
Add 135 to both sides:
4y = -9 + 135
4y = 126
Divide both sides by 4:
y = 31.5
So, the solution to the matrix equation is X = [-15; 31.5]
To find the solution of the matrix equation [9 4; 2 1]X = [-9 -6; -1 -8], we can use matrix operations.
First, let's represent the matrix equation in the standard form AX = B, where A is the coefficient matrix, X is the unknown matrix, and B is the constant matrix.
The coefficient matrix A is [9 4; 2 1], the unknown matrix X is [x y; z w], and the constant matrix B is [-9 -6; -1 -8].
To find X, we need to calculate the inverse of matrix A, denoted A^(-1).
The inverse of matrix A can be found by using the formula: A^(-1) = (1 / det(A)) * adj(A), where det(A) is the determinant of A and adj(A) is the adjugate of A.
First, let's calculate the determinant of matrix A:
det(A) = (9 * 1) - (4 * 2) = 9 - 8 = 1.
Next, let's calculate the adjugate of matrix A:
adj(A) = [1 -4; -2 9].
Now, we can calculate the inverse of matrix A:
A^(-1) = (1 / det(A)) * adj(A) = (1 / 1) * [1 -4; -2 9] = [1 -4; -2 9].
Now, we can find the unknown matrix X by multiplying both sides of the equation by A^(-1):
A^(-1) * A * X = A^(-1) * B.
Simplifying, we get:
X = A^(-1) * B = [1 -4; -2 9] * [-9 -6; -1 -8].
Performing matrix multiplication, we get:
X = [1 * (-9) + (-4) * (-1) 1 * (-6) + (-4) * (-8); -2 * (-9) + 9 * (-1) -2 * (-6) + 9 * (-8)].
Simplifying, we get:
X = [5 38; -7 -60].
Therefore, the solution to the matrix equation [9 4; 2 1]X = [-9 -6; -1 -8] is X = [5 38; -7 -60].
To find the solution of the matrix equation, we need to use matrix algebra.
Let's denote the unknown matrix as X:
X = [x1 x2; x3 x4]
The given matrix equation is:
[9 4; 2 1]X = [-9 -6; -1 -8]
Now, we can set up the equation using matrix multiplication:
[9*x1 + 4*x3 9*x2 + 4*x4] = [-9 -6]
[2*x1 + 1*x3 2*x2 + 1*x4] [-1 -8]
Let's solve for x1, x2, x3, and x4 using the system of equations:
9*x1 + 4*x3 = -9 ......(1)
9*x2 + 4*x4 = -6 ......(2)
2*x1 + 1*x3 = -1 ......(3)
2*x2 + 1*x4 = -8 ......(4)
To solve this system of equations, we can use various methods such as substitution or elimination. Let's use the elimination method here:
From equations (1) and (3), we can eliminate x1 by multiplying equation (3) by 4 and equation (1) by 2:
2*(2*x1) + 2*(1*x3) = 2*(-1)
4*x1 + 2*x3 = -2 ......(5)
Now, subtract equation (5) from equation (1):
(9*x1 + 4*x3) - (4*x1 + 2*x3) = (-9) - (-2)
5*x1 + 2*x3 = -7 ......(6)
From equations (2) and (4), we can eliminate x2 by multiplying equation (4) by 9 and equation (2) by 2:
2*(2*x2) + 2*(1*x4) = 2*(-8)
4*x2 + 2*x4 = -16 ......(7)
Now, subtract equation (7) from equation (2):
(9*x2 + 4*x4) - (4*x2 + 2*x4) = (-6) - (-16)
5*x2 + 2*x4 = 10 ......(8)
We now have a system of two equations with two variables:
5*x1 + 2*x3 = -7 ......(6)
5*x2 + 2*x4 = 10 ......(8)
To solve this system, we can use any method such as substitution or elimination. Let's use the substitution method here.
From equation (6), we can express x1 in terms of x3:
x1 = (-7 - 2*x3)/5 ......(9)
Now, substitute equation (9) into equation (8):
5*((-7 - 2*x3)/5) + 2*x3 = 10
Simplifying the equation:
-7 - 2*x3 + 2*x3 = 10
-7 = 10
The equation -7 = 10 is not true, which means there is no solution to the system of equations. Therefore, there is no solution to the given matrix equation [9 4; 2 1]X = [-9 -6; -1 -8].