If A = [[1, 2], [3, 4]], what is the inverse of A?
Practice Questions
1 question
Q1
If A = [[1, 2], [3, 4]], what is the inverse of A?
[[4, -2], [-3, 1]]
[[-2, 1], [1.5, -0.5]]
[[-2, 1], [1.5, -0.5]]
[[4, -2], [-3, 1]]
The inverse of A is (1/det(A)) * adj(A) = (1/(-2)) * [[4, -2], [-3, 1]] = [[-2, 1], [1.5, -0.5]].
Questions & Step-by-step Solutions
1 item
Q
Q: If A = [[1, 2], [3, 4]], what is the inverse of A?
Solution: The inverse of A is (1/det(A)) * adj(A) = (1/(-2)) * [[4, -2], [-3, 1]] = [[-2, 1], [1.5, -0.5]].
Steps: 6
Step 1: Identify the matrix A, which is A = [[1, 2], [3, 4]].
Step 2: Calculate the determinant of A (det(A)). The formula for the determinant of a 2x2 matrix [[a, b], [c, d]] is ad - bc. Here, a=1, b=2, c=3, d=4.
Step 3: Substitute the values into the determinant formula: det(A) = (1 * 4) - (2 * 3) = 4 - 6 = -2.
Step 4: Find the adjugate (adj) of A. The adjugate of a 2x2 matrix [[a, b], [c, d]] is [[d, -b], [-c, a]]. For our matrix, this gives us adj(A) = [[4, -2], [-3, 1]].
Step 5: Use the formula for the inverse of A, which is (1/det(A)) * adj(A). We already found det(A) = -2 and adj(A) = [[4, -2], [-3, 1]].
