Playbook
Determinants and Matrices
50 q - 1.12/paper - 48% HARD. Small and expensive. Adjoint, Determinant and the A adj(A) identity is 14 q at 64% HARD. The compensation is that its identities are memorisable and reusable, unlike most of the long tail.
- Questions in the bank
- 50
- q/paper (2024–25 shifts)
- 1.12
- Tagged HARD
- 48%
- Subtopics
- 3
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
50 q, 1.12/paper, 48% 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.
Adjoint, Determinant, and A·adj(A) Identity is the chapter's hardest corner at 64% HARD across 14 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.
System of Linear Equations and Symmetric Matrices (9 q, 44% HARD) is classification, not solving: 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.
Determinant evaluation and properties
Expansion along the sparsest row or column, plus the row and column operations that create zeros. Extracting a common factor from a row multiplies the determinant by that factor once, not n times.
The adjoint identities
A adj(A) = adj(A) A = |A| I; |adj A| = |A|^(n-1); adj(adj A) = |A|^(n-2) A for an invertible n by n matrix. Pure recall, and it converts several 64%-HARD questions into one-liners.
Inverse and its algebra
A inverse equals adj(A) divided by |A|, defined only when |A| is non-zero. Note the order reversal: (AB) inverse equals B inverse times A inverse.
Cayley-Hamilton and matrix polynomials
A matrix satisfies its own characteristic equation, which lets a high power of A be reduced to a linear combination of A and I. This is the standard route for A^n questions.
Consistency of a linear system
Compute the determinant of the coefficient matrix first. Non-zero means a unique solution; zero sends you to the numerator determinants to decide between inconsistent and infinitely many.
Determinant as a degeneracy test
Recognise the same 3x3-equals-zero condition when it appears as concurrency, collinearity, coplanarity or a scalar triple product. One computation, four chapter dialects.
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.
Drill every determinants and matrices question
50 questions from the bank, scoped to 3 bundled subtopics.
Drill one subtopic at a time
The 3 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.