If A = [[2, 3], [1, 4]], what is the inverse of A?
Practice Questions
1 question
Q1
If A = [[2, 3], [1, 4]], what is the inverse of A?
[[4, -3], [-1, 2]]
[[4, 3], [-1, 2]]
[[2, -3], [-1, 2]]
[[3, -2], [-1, 2]]
The inverse of A is given by (1/det(A)) * adj(A). Det(A) = (2*4) - (3*1) = 5. The adjoint is [[4, -3], [-1, 2]]. Thus, A^(-1) = (1/5) * [[4, -3], [-1, 2]].
Questions & Step-by-step Solutions
1 item
Q
Q: If A = [[2, 3], [1, 4]], what is the inverse of A?
Solution: The inverse of A is given by (1/det(A)) * adj(A). Det(A) = (2*4) - (3*1) = 5. The adjoint is [[4, -3], [-1, 2]]. Thus, A^(-1) = (1/5) * [[4, -3], [-1, 2]].
Steps: 6
Step 1: Identify the matrix A. Here, A = [[2, 3], [1, 4]].
Step 2: Calculate the determinant of A. Use the formula det(A) = (a*d) - (b*c), where A = [[a, b], [c, d]]. For our matrix, a = 2, b = 3, c = 1, d = 4. So, det(A) = (2*4) - (3*1) = 8 - 3 = 5.
Step 3: Find the adjoint (adjugate) of A. For a 2x2 matrix [[a, b], [c, d]], the adjoint is [[d, -b], [-c, a]]. For our matrix, the adjoint is [[4, -3], [-1, 2]].
Step 4: Use the formula for the inverse of A, which is A^(-1) = (1/det(A)) * adj(A). We already found det(A) = 5 and adj(A) = [[4, -3], [-1, 2]].
Step 5: Substitute the values into the formula: A^(-1) = (1/5) * [[4, -3], [-1, 2]].
Step 6: Multiply each element of the adjoint by 1/5 to get the final inverse: A^(-1) = [[4/5, -3/5], [-1/5, 2/5]].
