write the following simultaneous equations in the form of AX= B where A,X and B are matrices
11x+6y=6
9x+5y=7
hence write the solution for x and y as a product of two matrices.
To rewrite the simultaneous equations in the form AX = B, where A, X, and B are matrices, we first need to isolate the variables x and y on one side of the equation.
The system of equations can be written as:
11x + 6y = 6 ...(1)
9x + 5y = 7 ...(2)
We'll represent the variables x and y, as well as the constant terms on the right-hand side, as matrices:
A = [[11, 6], [9, 5]]
X = [[x], [y]]
B = [[6], [7]]
The matrix A contains the coefficients of the variables x and y, the matrix X contains the variables x and y, and the matrix B contains the constant terms.
Now, we can rewrite the original equations in matrix form as AX = B:
A * X = B
[[11, 6], [9, 5]] * [[x], [y]] = [[6], [7]]
To find the solution for x and y as a product of two matrices, we need to solve for X. We can do this by multiplying both sides of the equation by the inverse of matrix A.
X = A^(-1) * B
First, we need to find the inverse of matrix A.
The inverse of a 2x2 matrix can be found using the formula:
A^(-1) = (1/det(A)) * adj(A)
Where det(A) represents the determinant of matrix A, and adj(A) represents the adjugate of matrix A.
Let's start by finding det(A):
det(A) = (11 * 5) - (9 * 6) = 55 - 54 = 1
Next, we'll find the adjugate of matrix A by swapping the elements along the main diagonal and changing the sign of the off-diagonal elements:
adj(A) = [[5, -6], [-9, 11]]
Now, we can find the inverse of matrix A using the formula:
A^(-1) = (1/det(A)) * adj(A)
A^(-1) = (1/1) * [[5, -6], [-9, 11]] = [[5, -6], [-9, 11]]
Finally, we can find the solution for X by multiplying A^(-1) with B:
X = A^(-1) * B
[[5, -6], [-9, 11]] * [[6], [7]]
Evaluating this multiplication will give us the solution in the form of a matrix:
X = [[1], [2]]
Therefore, the solution for x and y is x = 1 and y = 2, represented as a product of two matrices.