All subjects · Further Mathematics · WFM03 Further Pure Mathematics 3

Transpose and 3 × 3 determinants

Edexcel IAL Further Mathematics · WFM03 Further Pure Mathematics 3

The determinant of a 3 × 3 matrix tells you whether it has an inverse and how it scales volume. The transpose swaps rows and columns, and it is needed for inverses and for orthogonal matrices.

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Transpose

The transpose A^T has the rows of A as its columns: (A^T)_ij = A_ji.

(AB)^T = B^T A^T (order reverses).

A matrix with A^T = A is symmetric.

Evaluating a 3 × 3 determinant

Expand along the top row, using the sign pattern + - +:

det A = a(ei - fh) - b(di - fg) + c(dh - eg) for rows (a, b, c), (d, e, f), (g, h, i).

Each 2 × 2 determinant is the minor of that entry. You can expand along any row or column using the checkerboard of signs; choose one with zeros to save work.

Singular matrices and properties

A is singular if det A = 0 (no inverse); otherwise non-singular.
det(AB) = det A × det B.
det(A^T) = det A.
For a 3 × 3 matrix, det(kA) = k³ det A.
|det A| is the volume scale factor; det A = 0 means space is squashed into a plane, line or point.

Worked example

Evaluate det A for A with rows (2, 1, 3), (0, -1, 4), (1, 2, 1).

2((-1)(1) - 4×2) = 2(-9) = -18
-1(0×1 - 4×1) = -1(-4) = 4
+3(0×2 - (-1)×1) = 3(1) = 3

Answer: det A = -18 + 4 + 3 = -11

Key idea: det A = a(ei - fh) - b(di - fg) + c(dh - eg). Singular means det = 0. det(AB) = det A det B; det(kA) = k³ det A; (AB)^T = B^T A^T.

Check yourself

Find the determinant of the matrix with rows (2, 1, 3), (0, -1, 4), (1, 2, 1).
Show answer
-11. 2(-1 - 8) - 1(0 - 4) + 3(0 + 1) = -18 + 4 + 3 = -11.
The matrix with rows (1, 2, k), (3, -1, 2), (0, 1, 1) is singular. Find k.
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3. det = 1(-1 - 2) - 2(3 - 0) + k(3 - 0) = -3 - 6 + 3k = 3k - 9 = 0, so k = 3.
Which is equal to (AB)^T?
- A^T B^T
- AB
- B^T A^T
- BA
Show answer
B^T A^T. Transposing a product reverses the order, just like inverting a product.
det A = 3 and det B = -2, where A and B are 3 × 3 matrices. Find det(AB).
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-6. det(AB) = det A × det B = -6.
A is a 3 × 3 matrix with det A = 5. Find det(2A).
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40. Each of the three rows is multiplied by 2, so det(2A) = 2³ × 5 = 40.
A unit cube is transformed by the matrix with rows (2, 0, 1), (1, 3, 0), (0, 1, 1). Find the volume of the image.
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7. det = 2(3 - 0) - 0 + 1(1 - 0) = 7, so the volume is 7.
A has rows (1, 4, 7), (2, 5, 8), (3, 6, 9). What is the second row of A^T?
- (2, 5, 8)
- (7, 8, 9)
- (3, 6, 9)
- (4, 5, 6)
Show answer
(4, 5, 6). The second row of A^T is the second column of A: 4, 5, 6.

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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.