Find the inverse of the matrix A = [[1, 2], [3, 4]].
Practice Questions
Q1
Find the inverse of the matrix A = [[1, 2], [3, 4]].
[[4, -2]; [-3, 1]]
[[1, -2]; [-3, 4]]
[[-2, 1]; [3, 4]]
[[2, -1]; [-1.5, 0.5]]
Questions & Step-by-Step Solutions
Find the inverse of the matrix A = [[1, 2], [3, 4]].
Correct Answer: [[-2, 1], [1.5, -0.5]]
Step 1: Identify the matrix A. Here, A = [[1, 2], [3, 4]].
Step 2: Calculate the determinant of A. The formula for the determinant of a 2x2 matrix [[a, b], [c, d]] is det(A) = ad - bc. For A, this is (1*4) - (2*3) = 4 - 6 = -2.
Step 3: Find the adjugate (adjoint) of A. For a 2x2 matrix, the adjugate is obtained by swapping the elements on the main diagonal and changing the signs of the off-diagonal elements. So, adj(A) = [[4, -2], [-3, 1]].
Step 4: Use the formula for the inverse of A, which is A^(-1) = (1/det(A)) * adj(A). We already found det(A) = -2 and adj(A) = [[4, -2], [-3, 1]].
Step 5: Substitute the values into the formula: A^(-1) = (1/-2) * [[4, -2], [-3, 1]].
Step 6: Multiply each element of adj(A) by (1/-2): A^(-1) = [[4*(-1/2), -2*(-1/2)], [-3*(-1/2), 1*(-1/2)]] = [[-2, 1], [1.5, -0.5]].
Matrix Inversion – The process of finding a matrix that, when multiplied with the original matrix, yields the identity matrix.
Determinant Calculation – The determinant is a scalar value that can be computed from the elements of a square matrix and is used to determine if the matrix is invertible.
Adjugate Matrix – The adjugate (or adjoint) of a matrix is the transpose of the cofactor matrix and is used in the calculation of the inverse.
