NDA Maths · Matrices & Determinants

Special Determinants: Trig, Complex, ω, Polynomial

Determinants whose entries are trig functions, complex numbers, cube roots of unity, or polynomial/sequence terms — each family has an identity that collapses it, very often to 0.

All Matrices & Determinants notes NDA Maths strategy

Why this matters

Twenty PYQs and the joint-hardest area in the chapter (50% HARD). These look intimidating but reward pattern recognition: a trig identity, ω's relation 1 + ω + ω² = 0, the powers of i, or an AP/GP row that forces two rows to be dependent. The four families below cover them — and the answer is 0 far more often than you'd expect.

Concept 1 of 4

Trigonometric determinants

Intuition

When entries are trig functions, the move is to apply an identity (Pythagorean, sum-to-product, or a triangle relation

A+B+C = \pi

) so that two rows/columns become equal or proportional — and the determinant drops to 0.

Definition

Use $\sin^2\theta + \cos^2\theta = 1$ , double-angle, and (for triangle problems) $A + B + C = \pi$ . Many such determinants are identically 0 because a trig identity makes rows dependent. Expand only after simplifying with the identity.

Worked example

Evaluate

\begin{vmatrix}\sin\theta & \cos\theta\\ -\cos\theta & \sin\theta\end{vmatrix}

$= (\sin\theta)(\sin\theta) - (\cos\theta)(-\cos\theta) = \sin^2\theta + \cos^2\theta$ .
By the Pythagorean identity this is $1$ for all $\theta$ .

Answer:

1

Practice this conceptself-check · 4 quick reps

Try it yourself

In triangle

ABC

, evaluate the determinant whose rows force a triangle identity such that two rows coincide. What is the typical value?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
$\cos^2\theta + \sin^2\theta = ?$
2.
$\cos^2 x - \sin^2 x = ?$
3.
First move on a trig determinant?
4.
In a triangle, $A + B + C = ?$

From the bank · past-year question

Example 1Matrices & DeterminantsEASY

The value of the determinant

\begin{vmatrix}\cos^2\dfrac{\theta}{2} & \sin^2\dfrac{\theta}{2} \\ \sin^2\dfrac{\theta}{2} & \cos^2\dfrac{\theta}{2}\end{vmatrix}

for all values of

\theta

[Q16 · Sep · 2017]

Drill 6 more on trigonometric determinants

Concept 2 of 4

Determinants with complex entries

Intuition

Treat

i

like any algebraic symbol but reduce its powers with the cycle

i, -1, -i, 1

(period 4). Expand normally; then collect real and imaginary parts to match a target

A + iB

Definition

Powers of $i$ : $i^1 = i,\ i^2 = -1,\ i^3 = -i,\ i^4 = 1$ , repeating every 4. After expanding a complex determinant, write it as $A + iB$ and read off $A$ (real) and $B$ (imaginary), or solve for unknowns by equating real/imaginary parts.

Powers of i (period 4)

i = i,\quad i^2 = -1,\quad i^3 = -i,\quad i^4 = 1

Worked example

Evaluate

\begin{vmatrix} 1+i & 1-i \\ 1-i & 1+i\end{vmatrix}

where

i = \sqrt{-1}

$= (1+i)^2 - (1-i)^2$ .
$(1+i)^2 = 2i$ and $(1-i)^2 = -2i$ .
So the determinant $= 2i - (-2i) = 4i$ .

Answer:

4i

Practice this conceptself-check · 4 quick reps

Try it yourself

If a complex determinant evaluates to

6 + 11i

and you must find real unknowns

x, y

inside it, what's the method?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
$i^2 = ?$
2.
$i^{15} = ?$
3.
$i^{12} = ?$
4.
To match $A + iB$ , you equate?

From the bank · past-year question

Example 2Matrices & DeterminantsHARD

What is the value of the determinant

\begin{vmatrix}i & i^2 & i^3 \\ i^4 & i^6 & i^8 \\ i^9 & i^{12} & i^{15}\end{vmatrix}

where

i=\sqrt{-1}

[Q12 · Apr · 2020]

Drill 2 more on determinants with complex entries

Concept 3 of 4

Cube-root-of-unity determinants

Intuition

The non-real cube root

\omega

obeys two relations that crush these determinants:

\omega^3 = 1

and

1 + \omega + \omega^2 = 0

. Substitute to reduce powers, then the sum-to-zero relation usually makes a row vanish.

Definition

For a non-real cube root of unity $\omega$ : $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$ . Reduce every power of $\omega$ mod 3, then use the sum relation — a row or column summing to $1 + \omega + \omega^2$ becomes 0, forcing the determinant to 0.

Cube roots of unity

\omega^3 = 1, \qquad 1 + \omega + \omega^2 = 0

Worked example

\omega = -\tfrac12 + i\tfrac{\sqrt3}{2}

, evaluate

\begin{vmatrix}1+\omega & 1+\omega^2\\ \omega & \omega^2\end{vmatrix}

Use $1 + \omega = -\omega^2$ and $1 + \omega^2 = -\omega$ (from $1+\omega+\omega^2=0$ ).
Determinant $= (-\omega^2)(\omega^2) - (-\omega)(\omega) = -\omega^4 + \omega^2$ .
$\omega^4 = \omega^3\cdot\omega = \omega$ , so $= -\omega + \omega^2 = \omega^2 - \omega$ .

Answer:

\omega^2 - \omega

(equivalently

-i\sqrt3

Practice this concept4 quick reps

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
$1 + \omega + \omega^2 = ?$
2.
$\omega^3 = ?$
3.
$1 + \omega = ?$
4.
$\omega^4 = ?$

From the bank · past-year question

Example 3Matrices & DeterminantsMODERATE

\omega

is a non-real cube root of unity, then what is a root of the following equation?

\begin{vmatrix} x+1 & \omega & \omega^2 \\ \omega & x+\omega^2 & 1 \\ \omega^2 & 1 & x+\omega \end{vmatrix} = 0

[Q12 · Apr · 2025]

Drill 2 more on cube-root-of-unity determinants

Concept 4 of 4

Polynomial and progression determinants

Intuition

If the rows are terms of an AP or GP (or shifted copies), they're linearly dependent and the determinant is 0. When a determinant is set equal to a polynomial

ax^4 + \dots

, match powers of

x

(or use the degree) to read off a coefficient.

Definition

AP/GP rows: three rows in arithmetic progression satisfy $R_1 + R_3 = 2R_2$ (dependent) → determinant 0; GP rows are proportional after a log/ratio step → 0. Determinant as polynomial: expand to a polynomial in $x$ and equate coefficients, or argue the degree to find a specific coefficient.

Worked example

Evaluate

\begin{vmatrix}1 & 1 & 1\\1 & 2 & 3\\1 & 4 & 9\end{vmatrix}

Subtract $C_1$ from $C_2$ and $C_3$ : $\begin{vmatrix}1&0&0\\1&1&2\\1&3&8\end{vmatrix}$ .
Expand along row 1: $1\cdot\begin{vmatrix}1&2\\3&8\end{vmatrix} = 1\cdot(8-6)$ .
$= 2$ . (This is a Vandermonde determinant in columns $1, k, k^2$ .)

Answer:

2

Practice this conceptself-check · 4 quick reps

Try it yourself

a, b, c

are in AP, what is

\begin{vmatrix}x+1&x+2&x+3\\x+2&x+3&x+4\\x+a&x+b&x+c\end{vmatrix}

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Three rows in AP ⇒ determinant?
2.
Rows of a GP (after ratio) are?
3.
To find a coefficient of $x^k$ in a determinant-polynomial?
4.
$R_1 + R_3 = 2R_2$ implies the rows are?

From the bank · past-year question

Example 4Matrices & DeterminantsMODERATE

What is the value of the determinant

\begin{vmatrix} 1! & 2! & 3! \\ 2! & 3! & 4! \\ 3! & 4! & 5! \end{vmatrix}

[Q11 · Sep · 2019]

Drill 5 more on polynomial and progression determinants

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Determinants with complex entries
Powers of i (period 4)
$i = i,\quad i^2 = -1,\quad i^3 = -i,\quad i^4 = 1$
Cube-root-of-unity determinants
Cube roots of unity
$\omega^3 = 1, \qquad 1 + \omega + \omega^2 = 0$

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Matrices & DeterminantsHARD

a

b

c

are the sides of a triangle

ABC

and

p

is the perimeter of the triangle, then what is

\begin{vmatrix}p+c & a & b\\c & p+a & b\\c & a & p+b\end{vmatrix}

equal to?

[Q19 · Sep · 2025]

Example 2Matrices & DeterminantsHARD

\begin{vmatrix} 2 & 3+i & -1 \\ 3 & -i & 0 \\ i & -1 & -1-i \\ 1 \end{vmatrix} = A + iB

where

i=\sqrt{-1}

, then what is

A+B

equal to?

[Q10 · Apr · 2025]

Example 3Matrices & DeterminantsHARD

\omega=-\dfrac{1}{2}+i\dfrac{\sqrt{3}}{2}

, then what is

\begin{vmatrix}1+\omega & 1+\omega^2 \\ \omega+\omega^2 & 1 \end{vmatrix} \cdot \begin{vmatrix}\omega & \omega^2 \\ 1 & \omega^2 \end{vmatrix} \cdot \begin{vmatrix}1 & \omega \\ 1 & \omega^2 \end{vmatrix}

equal to?

[Q12 · Sep · 2024]

Example 4Matrices & DeterminantsMODERATE

a

b

c

are in AP, then what is

\begin{vmatrix}x+1&x+2&x+3\\x+2&x+3&x+4\\x+a&x+b&x+3\end{vmatrix}

equal to?

[Q13 · Apr · 2023]

Example 5Matrices & DeterminantsMODERATE

A^2 + B^2 + C^2 = 0

, then what is the value of

\begin{vmatrix} 1 & \cos C & \cos B \\ \cos C & 1 & \cos A \\ \cos B & \cos A & 1 \end{vmatrix}

[Q11 · Apr · 2025]

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