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If D = [[4, 2], [1, 3]], what is the inverse of D?

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Question: If D = [[4, 2], [1, 3]], what is the inverse of D?

Options:

  1. [[3, -2], [-1, 4]]
  2. [[3, 2], [-1, 4]]
  3. [[4, -2], [-1, 3]]
  4. [[3, -4], [1, 2]]

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

Solution: 

The inverse of D is given by (1/det(D)) * adj(D). Here, det(D) = (4*3) - (2*1) = 10. The adjugate is [[3, -2], [-1, 4]]. Thus, D^(-1) = (1/10) * [[3, -2], [-1, 4]].

If D = [[4, 2], [1, 3]], what is the inverse of D?

Practice Questions

Q1
If D = [[4, 2], [1, 3]], what is the inverse of D?
  1. [[3, -2], [-1, 4]]
  2. [[3, 2], [-1, 4]]
  3. [[4, -2], [-1, 3]]
  4. [[3, -4], [1, 2]]

Questions & Step-by-Step Solutions

If D = [[4, 2], [1, 3]], what is the inverse of D?
  • Step 1: Identify the matrix D, which is given as D = [[4, 2], [1, 3]].
  • Step 2: Calculate the determinant of D using the formula det(D) = (a*d) - (b*c), where a, b, c, d are the elements of the matrix D. Here, a = 4, b = 2, c = 1, d = 3.
  • Step 3: Substitute the values into the determinant formula: det(D) = (4*3) - (2*1) = 12 - 2 = 10.
  • Step 4: Find the adjugate (adjoint) of D. The adjugate of a 2x2 matrix [[a, b], [c, d]] is given by [[d, -b], [-c, a]]. For D, this means adj(D) = [[3, -2], [-1, 4]].
  • Step 5: Use the formula for the inverse of D, which is D^(-1) = (1/det(D)) * adj(D). Substitute the values: D^(-1) = (1/10) * [[3, -2], [-1, 4]].
  • Step 6: Write the final result for the inverse of D: D^(-1) = [[3/10, -2/10], [-1/10, 4/10]].
  • Matrix Inversion – Understanding how to calculate the inverse of a 2x2 matrix using the formula involving the determinant and adjugate.
  • Determinant Calculation – Calculating the determinant of a 2x2 matrix correctly to ensure the inverse can be computed.
  • Adjugate Matrix – Finding the adjugate of a 2x2 matrix, which is essential for calculating the inverse.
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