Final answer:
The inverse of A is (3/2, -5/6). To solve the equations 3x + 4y = 1 and 5x + 2y = 3 using the inverse matrix, we can rewrite the equations in matrix form: AX = B, and the solution is X = (-4/3, 4/3). To find the inverse of the matrix A = (3/5 4/2), we can use the formula for the inverse of a 2x2 matrix.
Step-by-step explanation:
To find the inverse of the matrix A = (3/5 4/2), we can use the formula for the inverse of a 2x2 matrix. The inverse of A is given by:
A-1 = (1/ad - bc) * (d -b, -c/a)
where a, b, c, and d are the elements of the matrix A. Plugging in the values from the given matrix, we have:
A-1 = (1/(3/5 * 2/4 - 4/2 * 5/3)) * (2/4, -5/3) = (3/2, -5/6).
To solve the equations 3x + 4y = 1 and 5x + 2y = 3 using the inverse matrix, we can rewrite the equations in matrix form: AX = B, where X is the column vector (x, y) and B is the column vector (1, 3). The solution is then given by X = A-1 * B.
Using the values we calculated earlier for A-1 and B, we have:
X = (3/2, -5/6) * (1, 3) = (3/2 + (-5/6), -5/6 * 1 + 3 * 3) = (-4/3, 8/6) = (-4/3, 4/3).