All subjects · Further Mathematics · WFM01 Further Pure Mathematics 1

Determinants and inverse matrices

Edexcel IAL Further Mathematics · WFM01 Further Pure Mathematics 1

Every 2 × 2 matrix has a number called its determinant. If the determinant is not zero, the matrix has an inverse, which undoes its effect and lets you solve matrix equations.

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Determinant

For M = (a b; c d), det M = ad - bc.

If det M = 0, M is singular and has no inverse. If det M ≠ 0, M is non-singular.

Inverse

M-1 = (1/(ad - bc)) (d -b; -c a): swap a and d, change the signs of b and c, divide by the determinant.

MM-1 = M-1M = I. Check your inverse by multiplying.

For two non-singular matrices, (AB)-1 = B-1A-1: undo the last step first, like taking off shoes then socks.

Matrix equations

To solve MX = N, multiply on the left by M-1: X = M-1N.

To solve XM = N, multiply on the right: X = NM-1.

The side matters because matrix multiplication is not commutative.

Worked example

Find the inverse of M = (5 3; 2 1).

  1. det M = 5×1 - 3×2 = -1.
  2. Swap and change signs: (1 -3; -2 5).
  3. Divide by -1.

Answer: M-1 = (-1 3; 2 -5)

Key idea: det(a b; c d) = ad - bc; zero means singular. Inverse = (1/det)(d -b; -c a). (AB)-1 = B-1A-1.

Check yourself

  1. Find the determinant of (4 -2; 3 1).
    Show answer

    10. 4×1 - (-2)×3 = 4 + 6 = 10.

  2. Which matrix is singular?
    • (4 -2; 3 1)
    • (5 3; 2 1)
    • (2 6; 3 9)
    • (1 0; 0 1)
    Show answer

    (2 6; 3 9). det (2 6; 3 9) = 18 - 18 = 0.

  3. Find the values of k for which (k 4; 3 k - 4) is singular.
    Show answer

    6, -2. k(k - 4) - 12 = 0 gives k2 - 4k - 12 = (k - 6)(k + 2) = 0.

  4. Find the top-left entry of the inverse of (4 -2; 3 1). Give your answer as a fraction in its simplest form.
    Show answer

    1/10. M-1 = (1/10)(1 2; -3 4), so the top-left entry is 1/10.

  5. M = (4 -2; 3 1) and MX = (8 2; 11 5). Find the top-left entry of X.
    Show answer

    3. X = M-1(8 2; 11 5) = (1/10)(1 2; -3 4)(8 2; 11 5) = (3 6/5; 2 7/5).

  6. A and B are non-singular 2 × 2 matrices. Which equals (AB)-1?
    • A-1B-1
    • BA
    • (BA)-1
    • B-1A-1
    Show answer

    B-1A-1. (AB)(B-1A-1) = A(BB-1)A-1 = AA-1 = I.

  7. M is non-singular and XM = N. Which gives X?
    • X = M-1N
    • X = NM-1
    • X = N/M
    • X = MN-1
    Show answer

    X = NM-1. Multiply both sides on the right by M-1: XMM-1 = NM-1, so X = NM-1.

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Last updated 8 November 2026. Spotted a mistake? Tell us.