For the matrix D = [[4, 2], [1, 3]], find the inverse of D. (2022)
Practice Questions
Q1
For the matrix D = [[4, 2], [1, 3]], find the inverse of D. (2022)
[[3, -2], [-1, 4]]
[[3, 2], [-1, 4]]
[[3, -2], [1, 4]]
[[4, -2], [-1, 3]]
Questions & Step-by-Step Solutions
For the matrix D = [[4, 2], [1, 3]], find the inverse of D. (2022)
Step 1: Identify the matrix D. Here, D = [[4, 2], [1, 3]].
Step 2: Calculate the determinant of D. Use the formula det(D) = (a*d) - (b*c), where a, b, c, d are the elements of the matrix D = [[a, b], [c, d]]. For D, a = 4, b = 2, c = 1, d = 3. So, det(D) = (4*3) - (2*1) = 12 - 2 = 10.
Step 3: 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 4: Use the formula for the inverse of D, which is D^(-1) = (1/det(D)) * adj(D). We already found det(D) = 10 and adj(D) = [[3, -2], [-1, 4]].
Step 5: Substitute the values into the formula: D^(-1) = (1/10) * [[3, -2], [-1, 4]].
Step 6: Write the final answer: 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 as a prerequisite for finding the inverse.
Adjugate Matrix – Finding the adjugate of a 2x2 matrix, which is necessary for computing the inverse.
