Find the inverse of the matrix D = [[4, 7], [2, 6]]. (2023)

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Question: Find the inverse of the matrix D = [[4, 7], [2, 6]]. (2023)

Options:

Correct Answer: [[3/2, -7/4], [-1/2, 2/4]]

Exam Year: 2023

Solution:

The inverse of D is (1/det(D)) * adj(D) = (1/(4*6 - 7*2)) * [[6, -7], [-2, 4]] = [[3/2, -7/4], [-1/2, 2/4]].

Find the inverse of the matrix D = [[4, 7], [2, 6]]. (2023)

Find the inverse of the matrix D = [[4, 7], [2, 6]]. (2023)

Step 1: Identify the matrix D. Here, D = [[4, 7], [2, 6]].
Step 2: Calculate the determinant of D. The formula for the determinant of a 2x2 matrix [[a, b], [c, d]] is (a*d - b*c). For D, this is (4*6 - 7*2).
Step 3: Compute the determinant: 4*6 = 24 and 7*2 = 14. So, det(D) = 24 - 14 = 10.
Step 4: Find the adjugate of D. The adjugate of a 2x2 matrix [[a, b], [c, d]] is [[d, -b], [-c, a]]. For D, this is [[6, -7], [-2, 4]].
Step 5: Use the formula for the inverse of D, which is (1/det(D)) * adj(D). We already found det(D) = 10 and adj(D) = [[6, -7], [-2, 4]].
Step 6: Calculate the inverse: (1/10) * [[6, -7], [-2, 4]]. This means we multiply each element of the adjugate by 1/10.
Step 7: Perform the multiplication: [[6/10, -7/10], [-2/10, 4/10]] = [[3/5, -7/10], [-1/5, 2/5]].
Step 8: The final result is the inverse of D, which is [[3/5, -7/10], [-1/5, 2/5]].

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 formula for finding the inverse.