Properties of determinants — all 10 properties with examples and applications

Question

State all 10 properties of determinants and show how each property helps simplify determinant evaluation. Demonstrate with a $3 \times 3$ determinant.

(CBSE 12 Board + JEE Main pattern)

Solution — Step by Step

Here is the complete list. We will use a determinant $D$ of matrix $A$ of order $n$ :

#	Property	What it means
1	Transpose	$
2	Row swap	Swapping two rows changes sign: $D \to -D$
3	Identical rows	If two rows are identical, $D = 0$
4	Scalar multiple of row	Factor out $k$ from one row: $D' = kD$
5	Sum property	If a row is sum of two sets of elements, $D$ splits into sum of two determinants
6	Row operation (add multiple)	$R_i \to R_i + kR_j$ does not change $D$
7	All zeros in a row	$D = 0$
8	Proportional rows	If $R_i = kR_j$ , then $D = 0$
9	Scalar multiplication of matrix	$
10	Product property	$

Consider:

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

Apply $R_2 \to R_2 - R_1$ and $R_3 \to R_3 - R_1$ (Property 6 — value unchanged):

D = \begin{vmatrix} 1 & 2 & 3 \\ 3 & 3 & 3 \\ 6 & 6 & 6 \end{vmatrix}

Now $R_3 = 2R_2$ (proportional rows — Property 8), so $D = 0$ .

Suppose we need $|3A|$ where $A$ is $3 \times 3$ and $|A| = 5$ .

By Property 9: $|3A| = 3^3 \cdot |A| = 27 \times 5 = \mathbf{135}$

Students who write $|3A| = 3|A| = 15$ lose marks every year. The power is $n$ , not 1.

flowchart TD
    A["Given a Determinant"] --> B{"Any row/col all zeros?"}
    B -- Yes --> Z["D = 0"]
    B -- No --> C{"Two rows identical or proportional?"}
    C -- Yes --> Z
    C -- No --> D{"Can we simplify using R_i → R_i + kR_j?"}
    D -- Yes --> E["Apply row operations (D unchanged)"]
    E --> F{"Upper triangular form?"}
    F -- Yes --> G["D = product of diagonal"]
    F -- No --> D
    D -- No --> H["Expand along row/col with most zeros"]
    H --> I["Compute cofactors"]
    I --> J["Final Answer"]
    G --> J

Why This Works

Determinants encode the “signed volume” of the parallelepiped formed by row (or column) vectors. When two rows are identical, the parallelepiped collapses to a lower dimension — volume becomes zero. When we swap two rows, the orientation flips — hence the sign change.

Row operations of the type $R_i \to R_i + kR_j$ correspond to shearing, which preserves volume. That is why this operation does not change the determinant value, and it is our primary weapon for simplification.

The scalar multiplication property $|kA| = k^n|A|$ comes from the fact that each of the $n$ rows gets multiplied by $k$ , and Property 4 says each such multiplication pulls out one factor of $k$ .

Alternative Method

For small determinants, Sarrus’ Rule (only for $3 \times 3$ ) gives a quick computation without row operations:

D = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})

But for JEE, the row-reduction approach is faster and less error-prone for larger or parameter-heavy determinants.

In JEE Main, most determinant questions can be solved in under 2 minutes if you apply Property 6 first to create zeros, then expand along the row/column with the most zeros. Practice making the first column all zeros except one entry.

Common Mistake

The number one error: confusing $|kA|$ with $k|A|$ . For a $3 \times 3$ matrix, $|2A| = 8|A|$ , not $2|A|$ . The exponent equals the order of the matrix. This has appeared as a direct MCQ in CBSE boards and JEE Main multiple times — and students still get it wrong under exam pressure.