If D = [[4, 2], [1, 3]], find the inverse of D. (2022)

If D = [[4, 2], [1, 3]], find the inverse of D. (2022)

Step 1: Identify the matrix D, which is given as D = [[4, 2], [1, 3]].
Step 2: Calculate the determinant of D using the formula det(D) = (a*d) - (b*c), where D = [[a, b], [c, d]]. Here, a = 4, b = 2, c = 1, d = 3.
Step 3: Substitute the values into the determinant formula: det(D) = (4*3) - (2*1) = 12 - 2 = 10.
Step 4: Find the adjugate (adjoint) of D. The adjugate of a 2x2 matrix [[a, b], [c, d]] is given by [[d, -b], [-c, a]].
Step 5: Substitute the values into the adjugate formula: adj(D) = [[3, -2], [-1, 4]].
Step 6: Use the formula for the inverse of D, which is D^(-1) = (1/det(D)) * adj(D).
Step 7: Substitute the determinant and adjugate into the inverse formula: D^(-1) = (1/10) * [[3, -2], [-1, 4]].
Step 8: This gives the final result for the inverse of D.

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 the inverse.
Adjugate Matrix – Finding the adjugate of a 2x2 matrix, which is necessary for computing the inverse.