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Inverse of a 3 × 3 matrix

Edexcel IAL Further Mathematics · WFM03 Further Pure Mathematics 3

A non-singular 3 × 3 matrix has an inverse, found from its matrix of cofactors. The inverse undoes the transformation, so it takes an image back to the original point.

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Minors and cofactors

The minor of an entry is the 2 × 2 determinant left when you delete its row and column. The cofactor is the minor with the sign from the checkerboard

+ - +

The method

1. Find det A (stop if it is 0: no inverse).

2. Find the matrix of cofactors C.

3. Transpose it to get the adjugate, adj A = C^T.

4. A^-1 = (1/det A) adj A.

Check: A A^-1 = I.

Inverses of products and transformations

(AB)^-1 = B^-1A^-1 (order reverses).
If T maps x to Ax, then x = A^-1(image). Use this to find the point that maps to a given image.
The inverse of a rotation through θ is the rotation through -θ; for rotations and reflections the inverse equals the transpose. A reflection is its own inverse.
det(A^-1) = 1/det A.

Worked example

Find the inverse of A with rows (1, 2, 0), (0, 1, 3), (2, 0, 1).

det A = 1(1 - 0) - 2(0 - 6) + 0 = 13
Cofactors: rows (1, 6, -2), (-2, 1, 4), (6, -3, 1)
Transpose: adj A has rows (1, -2, 6), (6, 1, -3), (-2, 4, 1)

Answer: A^-1 = (1/13) × rows (1, -2, 6), (6, 1, -3), (-2, 4, 1)

Key idea: A^-1 = (1/det A) × transpose of the cofactor matrix. Remember the sign pattern. (AB)^-1 = B^-1A^-1. Use the inverse to undo a transformation.

Check yourself

For A with rows (1, 2, 0), (0, 1, 3), (2, 0, 1), find the cofactor of the entry in row 1, column 2.
Show answer
6. Delete row 1 and column 2: minor = 0×1 - 3×2 = -6. The sign in that position is -, so the cofactor is 6.
For the same matrix A, find the entry in row 1, column 3 of A^-1. Give your answer as a fraction in its simplest form.
Show answer
6/13. Entry (1, 3) of A^-1 is the cofactor of entry (3, 1) of A, divided by det A. That cofactor is +(2×3 - 0×1) = 6, so the entry is 6/13.
Which is equal to (AB)^-1?
- A^-1B^-1
- (BA)^-1
- B^-1A^-1
- A^-1 + B^-1
Show answer
B^-1A^-1. (AB)(B^-1A^-1) = A(BB^-1)A^-1 = AA^-1 = I.
The transformation with matrix A, rows (1, 2, 0), (0, 1, 3), (2, 0, 1), maps the point P to (3, 7, 4). Find the z-coordinate of P.
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2. P = A^-1(3, 7, 4) = (1/13)(3 - 14 + 24, 18 + 7 - 12, -6 + 28 + 4) = (1/13)(13, 13, 26) = (1, 1, 2).
R is the matrix of a rotation through θ about the z-axis. Which statement is true?
- R^-1 = -R
- R^-1 = R
- R has no inverse
- R^-1 = R^T, the rotation through -θ
Show answer
R^-1 = R^T, the rotation through -θ. Undoing a rotation means rotating back. For rotation matrices the columns are orthonormal, so R^T R = I.
A is a 3 × 3 matrix with det A = 13. Find det(A^-1). Give your answer as a fraction in its simplest form.
Show answer
1/13. det(A) det(A^-1) = det I = 1, so det(A^-1) = 1/13.
What should you check first when asked for the inverse of a 3 × 3 matrix?
- That it is symmetric
- That its trace is zero
- That all entries are positive
- That its determinant is non-zero
Show answer
That its determinant is non-zero. If det A = 0 the matrix is singular and has no inverse.

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Sources

Pearson Edexcel International Advanced Level in Mathematics, Further Mathematics and Pure Mathematics specification (issue 3), Unit FP3

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