JEE Mains Maths · Determinants

Simplifying Determinants by Row and Column Operations

Evaluating a determinant without brute expansion: taking out common factors, subtracting rows to create zeros, and spotting rows that make it vanish.

All Determinants notes JEE Mains Maths strategy

Why this matters

Ten PYQs, eight of them multiple choice. Five have rows or columns built from a pattern — an A.P., factorials, powers — that operations reduce quickly; five expand a simplified determinant and compare it with a given expression. Two ideas cover the page.

Concept 1 of 2: Factor out and create zeros

A factor common to a row or column comes outside. Subtracting one row from another does not change the determinant, so use it to create zeros before expanding. If one row equals a combination of others — for rows in A.P.,

R_1+R_3=2R_2

— the determinant is 0.

Definition

A common factor of a row or column comes outside.
$R_i\to R_i-kR_j$ leaves the determinant unchanged.
Two equal or proportional rows: the determinant is 0.
Rows in A.P.: $R_1-2R_2+R_3=0$ , so the determinant is 0.

Row operation

R_i\to R_i-kR_j:\ \Delta\ \text{unchanged}

Worked example

Evaluate

\begin{vmatrix}1&2&3\\4&5&6\\7&8&9\end{vmatrix}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2021 · Paper 21 · Q62Moderate

Example 1 · Determinants · Simplifying Determinants by Row and Column Operations

Let

A =\begin{bmatrix} \lbrack x + 1\rbrack & \lbrack x + 2\rbrack & \lbrack x + 3\rbrack \\ \lbrack x\rbrack & \lbrack x + 3\rbrack & \lbrack x + 3\rbrack \\ \lbrack x\rbrack & \lbrack x + 2\rbrack & \lbrack x + 4\rbrack \end{bmatrix}

, where

\lbrack t\rbrack

denotes the greatest integer less than or equal to

t

. If

det(A) = 192

, then the set of values of

x

is the interval:

Scaling a row scales the determinant

Replacing

R_i

2R_i-R_j

doubles the determinant. Only adding a multiple of another row is free.

Drill 4 more on factor out and create zeros

Concept 2 of 2: Expand and compare

After simplifying, expand along the row or column with the most zeros, and factor the result. Identities for symmetric determinants save the expansion: adding all columns of

\begin{vmatrix}x&1&1\\1&x&1\\1&1&x\end{vmatrix}

gives a common factor

x+2

. Compare the result with the expression in the question to read off the unknowns.

Definition

Expand along a row or column with zeros.
Symmetric: add all columns to one, then factor.
$\begin{vmatrix}1&a&a^2\\1&b&b^2\\1&c&c^2\end{vmatrix}=(a-b)(b-c)(c-a)$ .

Vandermonde

\begin{vmatrix}1&a&a^2\\1&b&b^2\\1&c&c^2\end{vmatrix}=(a-b)(b-c)(c-a)

Worked example

Factor

\begin{vmatrix}x&1&1\\1&x&1\\1&1&x\end{vmatrix}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 23 Jan 2026 Shift 1 · Q61Moderate

Example 2 · Determinants · Simplifying Determinants by Row and Column Operations

Among the statements : Statements (I) : If

\left| \begin{matrix} 1 & \cos\alpha & \cos\beta \\ \cos\alpha & 1 & \cos\gamma \\ \cos\beta & \cos\gamma & 1 \end{matrix} \right| = \left| \begin{matrix} 0 & \cos\alpha & \cos\beta \\ \cos\alpha & 0 & \cos\gamma \\ \cos\beta & \cos\gamma & 0 \end{matrix} \right|

, then

\cos^{2}\alpha + \cos^{2}\beta + \cos^{2}\gamma = \frac{3}{2}

, and Statements (II) : If

\left| \begin{matrix} x^{2} + x & x + 1 & x - 2 \\ 2x^{2} + 3x - 1 & 3x & 3x - 3 \\ x^{2} + 2x + 3 & 2x - 1 & 2x - 1 \end{matrix} \right|

= px + q

, then

p^{2} = 196q^{2}

Sign of the cofactor

Expanding along a row, the signs alternate

+,-,+

starting from the top-left. A middle-column term carries a minus sign.

Drill 4 more on expand and compare

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)

Factor out and create zeros
Row operation
$R_i\to R_i-kR_j:\ \Delta\ \text{unchanged}$
Expand and compare
Vandermonde
$\begin{vmatrix}1&a&a^2\\1&b&b^2\\1&c&c^2\end{vmatrix}=(a-b)(b-c)(c-a)$

Watch out for (2)

Scaling a row scales the determinant
→ Factor out and create zeros
Sign of the cofactor
→ Expand and compare

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 10 questions