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Matrices

High powers of a matrix by spotting a pattern, and the adjoint and determinant identities. Few long calculations.

Questions in the bank
115
q/paper in 2025–26
0.89
Numeric answer
38%
Notes pages
4

Tier: Long tail

When you’ll see it

A square matrix with a high power, an inverse or adjoint, a transpose condition, or a count of matrices with a given property.

How this chapter is tested

Powers of a matrix are the centre of the chapter. A matrix may repeat with a period, be I + N with N nilpotent, satisfy A² = A or A² = I, or satisfy its own characteristic equation by Cayley–Hamilton. In each case a high power reduces to a short expression, and the work is spotting the pattern before multiplying anything.

Adjoint and inverse questions are identity questions: |kA| = kⁿ|A|, A · adj A = |A| I, |adj A| = |A|ⁿ⁻¹ and adj(adj A) = |A|ⁿ⁻² A. They finish in a few lines of exponent arithmetic for anyone who knows them, and they overlap with Determinants. Symmetric, skew-symmetric and orthogonal matrices test the definitions and the split of any square matrix into a symmetric part plus a skew part.

The algebra page — multiplication, transpose, trace, counting matrices with given entries — is the base for all of it. Few questions need long entry-by-entry work. The costly errors carry rules from numbers into matrices: AB ≠ BA in general, AB = O does not force A = O or B = O, and (A + B)² is not A² + 2AB + B² unless AB = BA.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Matrix algebra

    Multiplication row by column, trace as the diagonal sum, (AB)ᵀ = BᵀAᵀ, and counting matrices whose entries meet a condition.

  • Powers and Cayley–Hamilton

    Find the period, write A = I + N, use A² = A or A² = I, or reduce with A² − (tr A)A + (det A)I = O for a 2 × 2 matrix.

  • Symmetric, skew-symmetric and orthogonal

    A = ½(A + Aᵀ) + ½(A − Aᵀ); a skew-symmetric matrix has a zero diagonal; AAᵀ = I gives det A = ±1.

  • Adjoint, inverse and determinant identities

    |kA| = kⁿ|A|, |adj A| = |A|ⁿ⁻¹, adj(adj A) = |A|ⁿ⁻² A, and A⁻¹ = adj A/|A| when |A| ≠ 0.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • kⁿ, not k

    For an n × n matrix, det(kA) = kⁿ det A. Options built on k det A are distractors.

  • The wrong exponent on the adjoint

    |adj A| = |A|ⁿ⁻¹, not |A|ⁿ; for a 3 × 3 matrix |adj(adj A)| = |A|⁴, not |A|².

  • Number rules in matrices

    AB ≠ BA in general, so (A + B)(A − B) is A² − B² only when A and B commute.

  • Orthogonal means det ±1

    AAᵀ = I gives (det A)² = 1, so det A can be −1 as well as +1.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Matrices notes

Drill every Matrices question

115 questions from the bank, across 4 subtopics.

Drill the 115 questions

Drill one subtopic at a time

The 4 subtopics, in teaching order.

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