Find the inverse of the matrix F = [[4, 7], [2, 6]]. (2021)

Find the inverse of the matrix F = [[4, 7], [2, 6]]. (2021)

Step 1: Identify the matrix F, which is F = [[4, 7], [2, 6]].
Step 2: Calculate the determinant of F using the formula det(F) = (4 * 6) - (7 * 2).
Step 3: Perform the multiplication: 4 * 6 = 24 and 7 * 2 = 14.
Step 4: Subtract the results: 24 - 14 = 10. So, det(F) = 10.
Step 5: Find the adjugate (adjoint) of F. The adjugate is formed by swapping the elements on the main diagonal and changing the signs of the off-diagonal elements.
Step 6: The main diagonal elements are 4 and 6. Swap them to get 6 and 4.
Step 7: The off-diagonal elements are 7 and 2. Change their signs to -7 and -2.
Step 8: So, adj(F) = [[6, -7], [-2, 4]].
Step 9: Now, use the formula for the inverse: F^(-1) = (1/det(F)) * adj(F).
Step 10: Substitute det(F) and adj(F) into the formula: F^(-1) = (1/10) * [[6, -7], [-2, 4]].
Step 11: Multiply each element of adj(F) by 1/10: F^(-1) = [[6/10, -7/10], [-2/10, 4/10]].
Step 12: Simplify the fractions: F^(-1) = [[3/5, -7/10], [-1/5, 2/5]].

Matrix Inversion – Understanding how to calculate the inverse of a 2x2 matrix using the formula involving the determinant and the adjugate.
Determinant Calculation – Calculating the determinant of a 2x2 matrix as a prerequisite for finding its inverse.
Adjugate Matrix – Finding the adjugate of a matrix, which is necessary for computing the inverse.