Properties Of Determinant

MathsDeterminantsFor JEE aspirants

The properties of determinants are shortcut rules that let you simplify or evaluate a determinant without expanding it directly. In JEE Mathematics, using properties like row-column interchange, scalar factoring, and the A.P. property can turn a lengthy $3 \times 3$ expansion into a single-line proof. This concept covers all nine core properties, along with their remarks and worked examples showing how to apply row and column operations to reduce a determinant to a triangular or zero form.

Key Formulas - Quick Reference

Transpose invariance: $det (A) = det (A^{T})$ . Rows and columns are interchangeable in all properties.
Row swap: swapping two rows (or columns) multiplies the determinant by $- 1$ .
Equal rows: if two rows (or columns) are identical or proportional, the determinant equals $0$ .
Scalar factor: multiplying one row (or column) by $k$ multiplies the determinant by $k$ .
Row combination: adding a multiple of one row (column) to another leaves the determinant unchanged.
Row splitting: if a row is a sum $R = R_{a} + R_{b}$ , the determinant splits: $det (R_{a} + R_{b}) = det (R_{a}) + det (R_{b})$ .
Zero row: if any row (or column) is entirely zero, the determinant is $0$ .
Factor theorem: if $D$ vanishes at $x = a$ , then $(x - a)$ is a factor of $D$ . If $r$ rows become identical at $x = a$ , then $(x - a)^{r - 1}$ is a factor.
A.P. property: if all rows (or columns) are in arithmetic progression, the determinant is $0$ .

The Nine Properties

Each property is stated for rows, but by Property 1 the same statement holds for columns.

Property 1: Transpose Invariance

If rows are changed into columns and columns into rows, the determinant remains unaltered:

det (A) = det (A^{T})

This is why every property in this list applies equally to rows and columns.

Property 2: Row/Column Swap

If any two rows (or columns) of a determinant are interchanged, the resulting determinant is the negative of the original:

det^{'} = - det

Remark: If a row of a determinant

D

is moved over

m

parallel rows (via

m

successive swaps), the resulting determinant

D^{'}

equals

(- 1)^{m} D

For example, if $D = a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3}$ , cycling the first row down two places gives $D^{'} = b_{1} b_{2} b_{3} c_{1} c_{2} c_{3} a_{1} a_{2} a_{3} = (- 1)^{2} D = D$ .

Property 3: Identical or Proportional Rows

If two rows (or two columns) have corresponding entries that are equal or proportional, the value of the determinant is

0

Property 4: Scalar Multiple of a Row

If each entry of one row (or column) of a determinant is multiplied by a nonzero constant

k

, then the determinant gets multiplied by

k

In formula form: $k a_{1} a_{2} a_{3} k b_{1} b_{2} b_{3} k c_{1} c_{2} c_{3} = k a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3}$ .

Property 5: Row Splitting

If each entry in a row (or column) of a determinant is written as the sum of two or more terms, then the determinant can be written as the sum of two or more determinants.

$a_{1} + p_{1} a_{2} + p_{2} a_{3} + p_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3} = a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3} + p_{1} p_{2} p_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3}$

Property 6: Row Combination (Elementary Operation)

If to each element of a row (or column) of a determinant we add the equimultiples of the corresponding elements of one or more parallel rows (columns), the determinant remains unaltered.

$a_{1} + l a_{2} + m a_{3} b_{1} + l b_{2} + m b_{3} c_{1} + l c_{2} + m c_{3} a_{2} b_{2} c_{2} a_{3} b_{3} c_{3} = a_{1} b_{1} c_{1} a_{2} b_{2} c_{2} a_{3} b_{3} c_{3}$

This is the most-used property in JEE. It lets you introduce zeros anywhere in a determinant without changing its value, dramatically simplifying the expansion.

Property 7: Zero Row

If each entry in any row (or column) of a determinant is zero, then the value of the determinant is

0

Property 8: Factor Theorem

If a determinant

D (x)

vanishes at

x = a

, then

(x - a)

is a factor of

D

. Equivalently, if two rows (or columns) become identical when

x = a

, then

(x - a)

is a factor of

D

For example, let $D = 1 a a^{2} 1 b b^{2} 1 c c^{2}$ . Substituting $a = b$ makes the first two columns identical, so $D = 0$ when $a = b$ , meaning $(a - b)$ is a factor of $D$ . Similarly $(b - c)$ and $(c - a)$ are factors, so this Vandermonde determinant equals $(a - b) (b - c) (c - a)$ .

Note: In general, if

r

rows (or columns) become identical when

a

is substituted for

x

, then

(x - a)^{r - 1}

is a factor of

D

Property 9: Arithmetic Progression

If in a determinant of order three or more, the elements in all the rows (or all the columns) are in A.P. - with the same or different common differences - then the value of the determinant is

0

This is a fast diagnostic: if every row (or column) is an A.P., stop expanding - the answer is zero.

Working Principles (Remarks)

All properties applicable to rows are also equally applicable to columns, but each row operation must be independent of column operations in a single step.
Whenever rows are disturbed by applications of properties, at least one row shall remain in its original shape. In other words, do not disturb all the rows at the same time.
It is always desirable to bring in as many zeros as possible in any row (or column) via Property 6 and then expand with respect to that row (column). Mere expansion from the outset should be avoided.
We can express a determinant as $∣ C_{1} C_{2} C_{3} ∣$ (in terms of its columns) or $R_{1} R_{2} R_{3}$ (in terms of its rows), where $C_{i}$ are columns and $R_{j}$ are rows.

Solved Example 1

Problem: Show that

Δ = 0

Δ = 11510106106197 97 - 8 88

Solution:

Operate $C_{2} \to C_{2} - \frac{1}{2} (C_{1} + C_{3})$ :

The new $C_{2}$ entries are: $106 - \frac{1}{2} (115 + 97) = 106 - 106 = 0$ ; $1 - \frac{1}{2} (10 + (- 8)) = 1 - 1 = 0$ ; $97 - \frac{1}{2} (106 + 88) = 97 - 97 = 0$ .

$Δ = 11510106000 97 - 8 88 = 0$

Note: Using Property 9 (A.P. property) one can immediately write

Δ = 0

directly, since each row is an A.P. - row 1:

115, 106, 97

(common difference

- 9

); row 2:

10, 1, - 8

(common difference

- 9

); row 3:

106, 97, 88

(common difference

- 9

Solved Example 2

Problem: Without expanding to any stage, prove that

Δ = 111 a b c a^{2} - b c b^{2} - c a c^{2} - ab = 0

Solution:

By Property 5 (row splitting), split column 3 across the two terms:

$Δ = 111 a b c a^{2} b^{2} c^{2} - 111 a b c b c c a ab = Δ_{1} - Δ_{2} (say)$

For $Δ_{2}$ , multiply $R_{1}$ by $a$ , $R_{2}$ by $b$ , $R_{3}$ by $c$ (each operation multiplies $Δ_{2}$ by the corresponding scalar, so we divide by $ab c$ at the end):

$Δ_{2} = \frac{1}{ab c} a b c a^{2} b^{2} c^{2} ab c ab c ab c = \frac{ab c}{ab c} a b c a^{2} b^{2} c^{2} 111$

Now swap $C_{1}$ and $C_{3}$ (a single swap), giving $111 a^{2} b^{2} c^{2} a b c \times (- 1)$ , then swap $C_{2}$ and $C_{3}$ (another swap, another $- 1$ ), giving $111 a b c a^{2} b^{2} c^{2}$ . Two sign flips cancel, so $Δ_{2} = Δ_{1}$ .

Hence $Δ = Δ_{1} - Δ_{1} = 0$ .

Solved Example 3

Problem: Evaluate

Δ = 1 lo g_{y} x lo g_{z} x lo g_{x} y 1 lo g_{z} y lo g_{x} z lo g_{y} z 1

, where

x, y, z

are positive.

Solution:

Using the change-of-base identity $lo g_{a} b = \frac{lo g b}{lo g a}$ , rewrite:

$Δ = 1 \frac{lo g x}{lo g y} \frac{lo g x}{lo g z} \frac{lo g y}{lo g x} 1 \frac{lo g y}{lo g z} \frac{lo g z}{lo g x} \frac{lo g z}{lo g y} 1$

Multiply $R_{1}$ , $R_{2}$ , $R_{3}$ by $lo g x$ , $lo g y$ , $lo g z$ respectively (this multiplies $Δ$ by $lo g x \cdot lo g y \cdot lo g z$ , but the resulting rows all become $(lo g x, lo g y, lo g z)$ ):

$(lo g x) (lo g y) (lo g z) \cdot Δ = lo g x lo g x lo g x lo g y lo g y lo g y lo g z lo g z lo g z = 0$

Since $x, y, z$ are positive and (assumed) not all equal to $1$ , the product $(lo g x) (lo g y) (lo g z)$ is generally nonzero, so $Δ = 0$ .

Common Mistakes to Avoid

Watch out

Disturbing all rows at once. When applying Property 6, at least one row must remain in its original form. Operations like " $R_{1} \to R_{1} - R_{2}$ and $R_{2} \to R_{2} - R_{1}$ " performed simultaneously are ambiguous - do them one at a time.
Confusing Property 4 (scalar row) with matrix scalar multiplication. For a determinant, multiplying one row by $k$ multiplies the determinant by $k$ . For a matrix, multiplying the whole matrix by $k$ multiplies its determinant by $k^{n}$ (where $n$ is the order).
Applying Property 5 incorrectly. You can split a row (or column) that is a sum, but you cannot split a product. $det (k A \cdot B) \neq = det (k A) + det (B)$ .
Missing the A.P. property. Before expanding, check whether every row (or every column) is in arithmetic progression - if so, the determinant is $0$ immediately, no work needed.
Forgetting the $(- 1)^{m}$ from cyclic rotations. Cycling a row over $m$ places is $m$ successive swaps, so the determinant is multiplied by $(- 1)^{m}$ , not left unchanged.
Applying properties inside the wrong bracket. If the determinant sits inside a larger expression like $2Δ + other terms$ , apply properties only to the determinant itself, not to the outer expression.

Frequently Asked Questions

Which property of determinants is used most often in JEE problems?

Property 6 (row-combination via elementary operations) is the workhorse. It lets you introduce zeros in a chosen row or column without changing the determinant's value, reducing a $3 \times 3$ expansion to a single $2 \times 2$ minor. A close second is Property 9 (A.P. property) for quick spot-diagnoses.

Does swapping two rows change the value of the determinant?

Swapping two rows (or two columns) multiplies the determinant by $- 1$ . If you swap twice, the sign returns to positive. In general, $m$ swaps multiply the determinant by $(- 1)^{m}$ .

Can I use both row and column operations in the same problem?

Yes. Row and column operations are independent, and both preserve the determinant's value when using Property 6. However, do not mix them in a single step - perform one row operation, note the result, then perform a column operation on the new determinant.

What is the factor theorem for determinants?

If a determinant $D (x)$ becomes zero when $x = a$ (typically because two rows or columns become identical), then $(x - a)$ is a factor of $D (x)$ . This is analogous to the factor theorem for polynomials and helps factor determinants like the Vandermonde without full expansion.

Why does an A.P. row make the determinant zero?

If every row is an A.P. with the same or different common difference, we can perform $R_{2} \to R_{2} - R_{1}$ and $R_{3} \to R_{3} - R_{2}$ ; the resulting rows differ only by their common differences, so two rows end up proportional (or one becomes zero). By Property 3 or 7, the determinant is $0$ .

Is $det (A + B) = det (A) + det (B)$ ?

No, this is false in general. Property 5 says a determinant splits when one row is a sum, keeping the other rows fixed. Adding two entire matrices is different from splitting a single row, and the row-splitting formula does not extend to full matrix addition.

How do I know which row operation to try first?

Look for common factors, near-equal rows, or arithmetic patterns. A useful heuristic: if two rows differ by a small amount, subtract them to create a row of small entries. If a column has one large entry, use row operations to zero out the other entries in that column, then expand along the column.

Do the properties change for a $4 \times 4$ or higher determinant?

No. All nine properties hold for square determinants of any order. Property 9 requires order at least three (an A.P. in a $2 \times 2$ does not force zero), but the rest apply universally, including for numerical computation of large determinants via row-reduction to triangular form.

Previous year questions on Properties Of Determinant

3 questions from past papers, each with a step-by-step solution.

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The Nine Properties

Property 1: Transpose Invariance

Property 2: Row/Column Swap

Property 3: Identical or Proportional Rows

Property 4: Scalar Multiple of a Row

Property 5: Row Splitting

Property 6: Row Combination (Elementary Operation)

Property 7: Zero Row

Property 8: Factor Theorem

Property 9: Arithmetic Progression

Working Principles (Remarks)

Common Mistakes to Avoid

Frequently Asked Questions

Which property of determinants is used most often in JEE problems?

Does swapping two rows change the value of the determinant?

Can I use both row and column operations in the same problem?

What is the factor theorem for determinants?

Why does an A.P. row make the determinant zero?

Is det(A+B)=det(A)+det(B)?

How do I know which row operation to try first?

Do the properties change for a 4×4 or higher determinant?

Previous year questions on Properties Of Determinant

Ready to master Determinants?

Is $det (A + B) = det (A) + det (B)$ ?

Do the properties change for a $4 \times 4$ or higher determinant?