What is the solution of the matrix equation?
[9 4] X= [-9 -6]
[2 1] [-1 -8]
To solve the matrix equation, we can use matrix multiplication.
Let's represent the matrix X as:
X = [a b]
[c d]
Multiplying the given matrix equation [9 4] X= [-9 -6] [2 1] [-1 -8] we get:
[9a + 4c, 9b + 4d] = [-9, -6]
[2a + c, 2b + d] = [-1, -8]
This gives us the following system of equations:
9a + 4c = -9
9b + 4d = -6
2a + c = -1
2b + d = -8
Solving these equations simultaneously, we find:
a = 1
b = -3
c = -1
d = -6
Therefore, the solution for matrix X is:
X = [1, -3]
[-1, -6]
To find the solution of the matrix equation [9 4] X = [-9 -6] [2 1] [-1 -8], we need to solve for the matrix X.
The equation can be written as AX = B, where:
A = [9 4] X = [x1 x2]
[2 1] [x3 x4]
B = [-9 -6]
[-1 -8]
To solve for X, we need to find the inverse of A and multiply it with B.
To find the inverse of matrix A, we apply the formula:
A^(-1) = 1 / (ad - bc) * [d -b]
[-c a]
Where a, b, c, and d are the elements of matrix A:
a = 9, b = 4, c = 2, d = 1
Calculating the determinant of A, we have:
ad - bc = (9 * 1) - (4 * 2) = 9 - 8 = 1
Using the formula for the inverse:
A^(-1) = 1/1 * [1 -4]
[-2 9]
Next, we multiply the inverse of A with B:
X = A^(-1) * B
Calculating the matrix product, we get:
X = [1 -4] * [-9 -6] = [-9*1 + 4*-1 -9*-4 + 4*-6] = [-13 42]
[-2 9] [-1 -8] [-2*1 + 9*-1 -2*-4 + 9*-6] [ 3 58]
Therefore, the solution to the matrix equation is:
x1 = -13, x2 = 42, x3 = 3, x4 = 58
So, the solution matrix X is:
[ -13 42]
[ 3 58]
To find the solution of the given matrix equation [9 4] X = [-9 -6] [2 1] [ -1 -8], we need to solve for the matrix X.
Let's denote X as [x y] (where x and y are unknown variables).
Now, we can set up the equation:
[9 4] * [x] = [-9]
[2 1] [y] [-6]
[-1]
[-8]
To solve this equation, we can use matrix inversion. The inverse of a 2x2 matrix [a b; c d] is given by:
1/(ad - bc) * [d -b; -c a]
Therefore, the inverse of the matrix [9 4; 2 1] is:
1/(9 * 1 - 4 * 2) * [1 -4; -2 9]
[9 -2]
[-4 1]
Now, we can multiply the inverse of the matrix [9 4; 2 1] by the given matrix [-9 -6] to obtain the solution matrix X:
[1 -4; -2 9] * [-9] = [x]
[-6] [y]
Calculating this, we get:
[(1 * -9) + (-4 * -6)] = [x]
[(-2 * -9) + (9 * -6)] [y]
Solving this, we get:
[15] = [x]
[-12] [y]
Therefore, the solution of the matrix equation is X = [15; -12].