Playbook
Determinants and Matrices
47 q - 1.13/paper - 47% HARD. Small and expensive. The determinant and adjoint identities page is 15 q at 67% HARD - the chapter's hardest corner and its most learnable, since three recalled identities answer most of it. The compensation is that its identities are memorisable and reusable, unlike most of the long tail.
- Questions in the bank
- 47
- q/paper (2024–25 shifts)
- 1.13
- Tagged HARD
- 47%
- Subtopics
- 4
Strand: Long Tail
When you’ll see it
A matrix raised to a power, an adjoint or an inverse asked for, or a 3x3 determinant set equal to zero.
How this chapter is tested
47 q, 1.13/paper, 47% HARD. Small and expensive — but it carries a compensation the rest of the tail does not: its content is a short list of identities that are memorisable and, unlike most tail material, reusable elsewhere on the paper.
Determinants, Cofactors and the Adjoint Identities is the chapter's hardest corner at 69% HARD across 16 q, and simultaneously its most learnable. Three lines answer most of it directly: A times adj(A) equals |A| times the identity, the determinant of adj(A) is |A| raised to (n - 1), and |kA| is k^n times |A| for an n by n matrix. Those are recall, not derivation.
The transferable idea is the vanishing determinant as a universal degeneracy test. A survey of the bank found it across five to six chapters and roughly 19 to 30 questions, surfacing as concurrency of three lines, collinearity of three points, coplanarity of two lines, the condition for a general second-degree equation to be a pair of lines, and the scalar triple product being zero. Learning to read 'determinant equals zero' as 'these objects are degenerate' pays well outside this chapter.
Systems of Linear Equations and Symmetric, Skew-Symmetric Matrices (8 q, 38% HARD) is half solving and half classification: a non-zero determinant means a unique solution, a zero determinant means either no solution or infinitely many, and telling those two apart is the whole question.
The sub-skills
The distinct skills inside the chapter, in the order to learn them.
Determinants, Cofactors and the Adjoint Identities
Expansion along a row with cofactors (an alien expansion gives zero), the adjoint as the transposed cofactor matrix, and three recalled identities — A adj(A) = |A| I, |adj A| = |A|^(n-1), |kA| = k^n |A| — that convert the chapter's 69%-HARD corner into one-liners. The A adj(A) = A A^T stem is two equations, one from the off-diagonal and one from the diagonal.
Inverse of a Matrix — Adjoint Formula, Products and Verification
A inverse equals adj(A) divided by |A|, defined only when |A| is non-zero; for an expression like A^2 - 5A or A + B, form the matrix first, then invert. Note the order reversal: (AB) inverse equals B inverse times A inverse, so B inverse = (AB) inverse times A. Unknown entries come from A A inverse = I.
Cayley–Hamilton, Matrix Polynomials and Powers
A 2x2 matrix satisfies A^2 - (trace) A + |A| I = 0, so A inverse = (trace I - A)/|A| gives alpha and beta on sight, a factored polynomial in A gives A inverse in one line, and a high power of A reduces through the cycle at which A^m returns to a scalar times I.
Systems of Linear Equations and Symmetric, Skew-Symmetric Matrices
Solve AX = B by elimination when the coefficients are small integers; a homogeneous system has non-trivial solutions exactly when the determinant vanishes; any square matrix splits into (M + M^T)/2 plus (M - M^T)/2, and an odd-order skew-symmetric matrix is singular.
Traps to expect
Distractor shapes this chapter reuses. The Traps page covers the patterns that cut across chapters.
Scalar multiple of a matrix
|kA| is k^n |A| for an n by n matrix, not k |A|. For 3x3 that is a factor of k^3, and the option built on k |A| is always present.
Determinant of the adjoint
|adj A| is |A|^(n-1), not |A|. For 3x3 that squares the determinant, so the wrong option is the un-squared value.
Assuming commutativity
AB is not BA in general, so (AB) inverse is B inverse A inverse and (AB) transpose is B transpose A transpose. The un-reversed order is the distractor.
Zero determinant read as no solution
A singular system may still have infinitely many solutions. An option asserting 'no solution' on the strength of |A| = 0 alone is the trap.
Learn it before you drill it
This chapter has full teaching notes — foundations, worked examples, self-checks and a per-subtopic mastery checkpoint. Read the notes once, then drill subtopic by subtopic below.
Determinants and Matrices notesDrill every determinants and matrices question
47 questions from the bank, scoped to 4 bundled subtopics.
Drill one subtopic at a time
The 4 subtopics this playbook covers, in catalog order.
Related playbooks
Often paired with this one — the technique, the trap or the taxonomy overlaps. Drill these next.