What is the solution of the matrix equation?
[9 4];[2 1]X=[-9 -6];[-1 -8]
(Answer is a matrix)
The solution is the matrix:
[-1 -2]
[2 0]
To find the solution of the matrix equation [9 4; 2 1] X = [-9 -6; -1 -8], we can use matrix algebra.
Let's call the unknown matrix X as [a b; c d]. So, our equation becomes:
[9 4; 2 1] [a b; c d] = [-9 -6; -1 -8]
Now, we can multiply the two matrices together and set each element in the resulting matrix equal to the corresponding element in the right-hand side matrix.
Multiplying the matrices, we get:
[9a + 4c 9b + 4d; 2a + c 2b + d] = [-9 -6; -1 -8]
Now, we can equate the corresponding elements:
9a + 4c = -9
9b + 4d = -6
2a + c = -1
2b + d = -8
To solve this system of equations, we can use any method such as substitution, elimination, or matrix inverse. Let's use the matrix inverse method.
The system of equations can be written in matrix form as:
[9 4; 2 1] [a b; c d] = [-9 -6; -1 -8]
To find the solution for the matrix X, we can multiply both sides of the equation by the inverse of [9 4; 2 1]:
[9 4; 2 1]^-1 [9 4; 2 1] [a b; c d] = [9 4; 2 1]^-1 [-9 -6; -1 -8]
The inverse of the matrix [9 4; 2 1] can be calculated using various methods such as Gauss-Jordan elimination or computing the adjugate matrix divided by the determinant. Once the inverse is obtained, we can proceed with matrix multiplication.
After calculating the inverse and multiplying the matrices, we get:
[a b; c d] = [2 -4; -2 9]
Therefore, the solution to the matrix equation is:
X = [a b; c d] = [2 -4; -2 9]
To find the solution of the matrix equation [9 4; 2 1]X = [-9 -6; -1 -8], we need to first find the inverse of the coefficient matrix [9 4; 2 1].
To find the inverse of a 2x2 matrix [a b; c d], we can use the formula:
1 / (ad - bc) * [d -b; -c a]
In this case, the coefficient matrix is [9 4; 2 1].
The determinant (ad - bc) of this matrix is (9 * 1) - (4 * 2) = 9 - 8 = 1.
Therefore, the inverse of the coefficient matrix is 1 / 1 * [1 -4; -2 9].
Next, we need to multiply the inverse of the coefficient matrix by the constant matrix:
1 / 1 * [1 -4; -2 9] * [-9 -6; -1 -8]
Performing the matrix multiplication:
[1 * -9 + -4 * -1 1 * -6 + -4 * -8]
[-2 * -9 + 9 * -1 -2 * -6 + 9 * -8]
Simplifying further:
[9 + 4 6 + 32]
[18 - 9 12 - 72]
[13 38]
[9 - 60]
Therefore, the solution to the matrix equation [9 4; 2 1]X = [-9 -6; -1 -8] is:
X = [13 38; 9 - 60]