What is the inverse of the matrix A = [[1, 2], [3, 4]]?
Practice Questions
Q1
What is the inverse of the matrix A = [[1, 2], [3, 4]]?
[[4, -2], [-3, 1]]
[[-2, 1], [1.5, -0.5]]
[[-2, 1], [1.5, -0.5]]
[[2, -1], [-1.5, 0.5]]
Questions & Step-by-Step Solutions
What is 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, a=1, b=2, c=3, d=4. So, det(A) = (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 a and d, and changing the signs of b and c. 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]].
