JEE Mains Maths · Determinants

Determinants of kA, adj A and Products

The rules for the determinant of a scalar multiple, an adjoint, an inverse and a product, and how they chain through nested expressions.

All Determinants notes JEE Mains Maths strategy

Why this matters

Twenty-nine PYQs, twenty of them multiple choice, nine numerical. Eighteen chain the rules for |kA|, |adj A| and nested adjoints; eleven use products, inverses and cofactor matrices. Each is a few exponents once the rules are right, and the order n of the matrix is the usual slip. Two ideas cover the page.

Concept 1 of 2: Scalar multiples and adjoints

For an

n\times n

matrix, multiplying by

k

scales every row, so

|kA|=k^n|A|

. From

A\,\mathrm{adj}A=|A|I

come

|\mathrm{adj}A|=|A|^{n-1}

and

\mathrm{adj}(\mathrm{adj}A)=|A|^{n-2}A

. Work nested expressions from the inside out, one rule at a time.

Definition

$|kA|=k^n|A|$ .
$A\,\mathrm{adj}A=\mathrm{adj}A\,A=|A|I$ ; $|\mathrm{adj}A|=|A|^{n-1}$ .
$\mathrm{adj}(\mathrm{adj}A)=|A|^{n-2}A$ ; $|\mathrm{adj}(\mathrm{adj}A)|=|A|^{(n-1)^2}$ .
$\mathrm{adj}(kA)=k^{n-1}\mathrm{adj}A$ ; $A^{-1}=\frac{\mathrm{adj}A}{|A|}$ .

Adjoint rules

|kA|=k^n|A|,\qquad|\mathrm{adj}A|=|A|^{n-1}

Worked example

A

3\times3

with

|A|=2

. Find

|\mathrm{adj}(2A)|

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 22 Jan 2025 · Q73Moderate

Example 1 · Determinants · Determinants of kA, adj A and Products

Let

A

be a square matrix of order 3 such that

det(A) = - 2

and

det(3adj( - 6adj(3A))) =2^{m + n}\cdot3^{mn}

m > n

. Then

4\text{ }m + 2n

is equal to

\_\_\_\_\_\_\_

The power depends on the order

|kA|=k^n|A|

, not

k|A|

, and

|\mathrm{adj}A|=|A|^{n-1}

changes with

n

. Read the order of the matrix before applying either.

Drill 17 more on scalar multiples and adjoints

Concept 2 of 2: Products, inverses and cofactors

The determinant of a product is the product of determinants, so

|P^{-1}AP|=|A|

and

|A^m|=|A|^m

. The cofactor matrix is the transpose of the adjoint, so it has the same determinant. A row operation that adds a multiple of one row to another leaves the determinant unchanged; swapping two rows changes its sign.

Definition

$|AB|=|A||B|$ , $|A^{-1}|=\frac1{|A|}$ , $|A^T|=|A|$ .
$|P^{-1}AP|=|A|$ .
Cofactor matrix $C=(\mathrm{adj}A)^T$ : $|C|=|A|^{n-1}$ .
Row swap: sign changes; $R_i\to R_i+kR_j$ : unchanged.

Product rule

|AB|=|A|\,|B|

Worked example

A,B

are

3\times3

|A|=3

|B|=-2

. Find

|2AB^{-1}|

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2024 · 29 January 2024 · Q151Moderate

Example 2 · Determinants · Determinants of kA, adj A and Products

Let

A =\begin{bmatrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{bmatrix}

and

P =\begin{bmatrix} 1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5 \end{bmatrix}

. The sum of the prime factors of

\left| P^{- 1}AP - 2I \right|

is equal to

Divide by a negative determinant

When

|A|

is negative, formulas like

|X|=\frac{|AX|}{|A|}

flip the sign. Stopping at

|A||X|

gives the wrong sign.

Drill 10 more on products, inverses and cofactors

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)

Scalar multiples and adjoints
Adjoint rules
$|kA|=k^n|A|,\qquad|\mathrm{adj}A|=|A|^{n-1}$
Products, inverses and cofactors
Product rule
$|AB|=|A|\,|B|$

Watch out for (2)

The power depends on the order
→ Scalar multiples and adjoints
Divide by a negative determinant
→ Products, inverses and cofactors

Test yourself on Determinants

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Determinants chapter test Drill all 29 questions