If D = [[4, 2], [1, 3]], what is the inverse of D?
Practice Questions
Q1
If D = [[4, 2], [1, 3]], what is the inverse of D?
[[3, -2], [-1, 4]]
[[3, 2], [-1, 4]]
[[4, -2], [-1, 3]]
[[3, -4], [1, 2]]
Questions & Step-by-Step Solutions
If D = [[4, 2], [1, 3]], what is the inverse of D?
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 a, b, c, d are the elements of the matrix 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]]. For D, this means adj(D) = [[3, -2], [-1, 4]].
Step 5: Use the formula for the inverse of D, which is D^(-1) = (1/det(D)) * adj(D). Substitute the values: D^(-1) = (1/10) * [[3, -2], [-1, 4]].
Step 6: Write the final result for the inverse of D: D^(-1) = [[3/10, -2/10], [-1/10, 4/10]].
Matrix Inversion – Understanding how to calculate the inverse of a 2x2 matrix using the formula involving the determinant and adjugate.
Determinant Calculation – Calculating the determinant of a 2x2 matrix correctly to ensure the inverse can be computed.
Adjugate Matrix – Finding the adjugate of a 2x2 matrix, which is essential for calculating the inverse.
