medium difficulty7 min

Properties of Determinants

Properties of determinants are rules that allow simplification by operating on rows or columns without expanding fully.

In one line: Swap changes sign, equal makes zero, add multiple changes nothing.

Practice This Concept

Understand it

Interchanging two rows or two columns changes the sign of the determinant.
If two rows or columns are identical or proportional, the determinant is zero.
Multiplying one row or column by k multiplies the determinant by k.
Adding a multiple of one row or column to another does not change the determinant.
These properties are valid when applied consistently to entire rows or entire columns.

Example

If D = |[[1,2],[3,4]]| = -2, then interchanging the two rows gives |[[3,4],[1,2]]| = 2.

Formula and rule

R_i ↔ R_j changes D to -D.

If R_i = R_j or R_i = kR_j, then D = 0.

If R_i is multiplied by k, determinant becomes kD.

The operation R_i → R_i + kR_j does not change D.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Let D = |[[1,2,3],[1,2,3],[4,5,6]]|.

1Since R1 = R2, two rows are identical.
2Therefore D = 0 without expansion.

Method you can reuse

1Look for identical, proportional, or easily transformable rows or columns.
2Apply one property at a time.
3Track whether the determinant sign or factor changes.
4Stop when a zero row, triangular form, or simple expansion appears.

Visual tip

No diagram is needed. A small row-operation table may be useful for revision.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What is the value of a determinant if its first and third rows are identical?

Think first, then show answer

The determinant is 0 because two rows are identical.

Common wrong answer

Changing the sign after using R_i → R_i + kR_j, even though this operation does not change the determinant.

Don't mix this up

Multiplying only one element of a row and treating it as if the whole row was multiplied.

Exam tip

Mention the property used before simplifying, especially in long-answer determinant simplification questions.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Questions often ask students to prove a determinant identity, simplify a determinant containing variables, or evaluate a determinant quickly using row or column operations.

1
Enough for 1 mark
Properties of determinants are rules that allow simplification by operating on rows or columns without expanding fully.
2
Add for 2 marks
Example: If D = |[[1,2],[3,4]]| = -2, then interchanging the two rows gives |[[3,4],[1,2]]| = 2.
3
Add for 3 marks
Interchanging two rows or two columns changes the sign of the determinant.

MCQ Quiz

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