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Determinants

Mostly systems of three equations with a constant left open: which values give one, none or infinitely many solutions.

Questions in the bank
136
q/paper in 2025–26
0.84
Numeric answer
16%
Notes pages
8

Tier: Long tail

When you’ll see it

A system of three linear equations with a constant left open, or the determinant of kA, adj A or a matrix whose entries depend on x.

How this chapter is tested

Most of this chapter is not about determinants for their own sake. It is systems of three equations in three unknowns with one or two constants left open, and the question asks which values give infinitely many solutions, none, or exactly one. The work is Δ = 0 and one Cramer determinant, or spotting that one equation is a combination of the other two.

The order of checks is fixed. Δ ≠ 0 means exactly one solution. Δ = 0 with any of Δx, Δy, Δz non-zero means none. Δ = 0 with all three zero usually means infinitely many, but confirm by elimination. Time goes on the questions that list several statements about a system, and on counting the values of an angle in an interval that make Δ = 0.

The last pages are shorter: chains of |kA| and |adj A|, patterned determinants reduced by row and column operations, and determinants in x that are simplified, differentiated or integrated. The adjoint rules are shared with Matrices, and a determinant in θ set to zero is finished with the tools of Trigonometric Equations.

The sub-skills

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

  • Infinitely many solutions

    Write the third equation as a combination of the first two, or set Δ = 0 and one Cramer determinant to zero, and solve for both constants.

  • No solution

    Δ = 0 with at least one of Δx, Δy, Δz non-zero; test each root of Δ, because some give infinitely many instead.

  • Classifying a system

    Exactly one solution when Δ ≠ 0; test each statement in the options on its own.

  • Homogeneous systems

    A non-trivial solution exists exactly when Δ = 0; with an angle in the coefficients, solve that equation in the given interval.

  • Determinants of kA and adj A

    For an n × n matrix, |kA| = kⁿ|A|, |adj A| = |A| to the power n − 1, and |AB| = |A||B|.

  • Row and column operations

    Take out common factors and create zeros before expanding; rows built from an A.P., powers or factorials collapse quickly.

  • Determinants as functions of x

    Reduce to one expression, then find its range or roots; to differentiate, differentiate one row at a time and add.

Traps to expect

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

  • The order of the matrix

    |2A| for a 3 × 3 matrix is 8|A|, not 2|A|. Every power in the adjoint chain depends on n.

  • Δ = 0 is not yet infinitely many

    A root of Δ can leave the system with no solution. Check the Cramer determinants, or eliminate, before choosing.

  • Scaling one row

    Multiplying a single row by k multiplies the determinant by k, not by kⁿ.

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.

Determinants notes

Drill every Determinants question

136 questions from the bank, across 8 subtopics.

Drill the 136 questions

Drill one subtopic at a time

The 8 subtopics, in teaching order.

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