Determinants
The determinant is a single number measuring how a matrix scales volume, and it is zero exactly when the matrix is singular. For 2×2 it is ad − bc; for larger matrices, cofactor expansion along a row or column works, but row reduction to triangular form is faster since a triangular determinant is just the product of the diagonal.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Use ad − bc for 2×2The base case for every cofactor expansion.
- Expand along the row or column with most zerosEach zero entry kills a whole cofactor, so choosing well can halve the work.
- Get the sign pattern rightCofactor signs alternate in a checkerboard starting + at the top left: (−1)^(i+j).
- Or row reduce to triangularAdding a multiple of one row to another leaves the determinant unchanged; swapping rows negates it; scaling a row scales it.
Worked example
Compute the determinant of [[2, 0, 1], [3, −1, 2], [1, 4, 0]].
- Expand along the first row, which contains a zero.
- 2·det[[−1,2],[4,0]] − 0·(…) + 1·det[[3,−1],[1,4]].
- First 2×2: (−1)(0) − (2)(4) = −8. Third: (3)(4) − (−1)(1) = 13.
- Total: 2(−8) + 1(13) = −16 + 13.
Answer. −3. Non-zero, so the matrix is invertible.
Where marks get dropped
These are the specific errors that cost credit on determinants questions — QED's rubric penalises each of them separately.
- Getting the alternating signs wrong in cofactor expansion — the (2,1) cofactor carries a minus.
- Assuming det(A + B) = det A + det B. It is false in general; determinants are multiplicative, not additive.
- Forgetting that swapping two rows negates the determinant when using row reduction.
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Determinants — frequently asked questions
What does the determinant mean geometrically?
The signed volume scale factor of the linear map. A determinant of 0 means the map collapses space into a lower dimension, which is why it detects singularity.
Is det(AB) = det A · det B?
Yes, always. It follows that det(A⁻¹) = 1/det A and that similar matrices have equal determinants.
Which method is fastest?
For 2×2 and 3×3, direct expansion. For 4×4 and beyond, row reduce to triangular form — cofactor expansion costs O(n!) while elimination costs O(n³).
The rest of Linear Algebra
Matrices, systems, determinants, eigenvalues. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Matrix arithmetic & inverses
- 2Gaussian elimination
- 3Determinants
- 4Vector spaces, span & linear independence
- 5Basis & dimension
- 6Eigenvalues & eigenvectors
- 7Linear maps & their matrices
- 8Rank, nullity & the rank–nullity theorem
- 9Column space, null space & solution sets
- 10Orthogonality, projections & Gram–Schmidt
- 11Diagonalisation & powers of a matrix
- 12Dot product, norms & angles
- 13Change of basis & similar matrices
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