In ECET 2026, Mathematics is a high-scoring subject where even a small concept can give you an edge. Determinants form a key part of Linear Algebra and are often tested for their properties. By mastering these properties, you can solve problems quickly without lengthy calculations—perfect for time-pressured competitive exams like ECET.
📘 Concept Notes – Determinants & Their Properties
A determinant is a scalar value calculated from a square matrix. It is used in solving systems of linear equations, finding inverses of matrices, and in many applications of engineering mathematics.
✏️ Important Properties of Determinants
Let $A$ be a square matrix of order $n$, and $\det(A)$ be its determinant.
- Interchanging two rows (or columns) changes the sign
If two rows (or columns) are swapped, the determinant is multiplied by $-1$. - Identical rows (or columns) ⇒ Determinant = 0
If two rows (or columns) are the same, the determinant is zero. - If any row (or column) is multiplied by a constant $k$
Determinant is multiplied by $k$. - If all elements of a row (or column) are multiplied by the same constant and added to another row (or column)
Determinant remains unchanged. - Determinant of a triangular matrix
For upper or lower triangular matrices, the determinant = product of the diagonal elements. - Transposing a matrix doesn’t change its determinant
$\det(A) = \det(A^T)$. - If a row (or column) has all zeros
Determinant = 0. - Scalar multiplication rule
Multiplying the entire matrix by $k$ multiplies the determinant by $k^n$ (where $n$ is the order of the matrix).
💡 Quick Tip for Exams:
Instead of expanding determinants using cofactor expansion every time, use row/column operations with these properties to simplify calculations.
🔟 10 Most Expected MCQs – ECET 2026 [Determinants – Properties]
Q1. If two rows of a determinant are interchanged, the new determinant is:
A) Same as before
B) Double
C) Negative of the original
D) Zero
Q2. If a determinant has two identical rows, its value is:
A) Zero
B) One
C) Unchanged
D) Negative
Q3. If a row is multiplied by 5, the determinant is:
A) Multiplied by 5
B) Multiplied by 25
C) Multiplied by 125
D) Unchanged
Q4. The determinant of a matrix and its transpose are:
A) Always equal
B) Always opposite
C) Sometimes equal
D) Always zero
Q5. If one row of a determinant has all zero elements, its value is:
A) 1
B) 0
C) -1
D) Cannot be determined
Q6. Determinant of a triangular matrix is equal to:
A) Sum of diagonal elements
B) Product of diagonal elements
C) Sum of all elements
D) Zero
Q7. If all rows are multiplied by the same constant $k$, determinant is multiplied by:
A) $k$
B) $k^n$
C) $k^{n-1}$
D) 0
Q8. Adding a multiple of one row to another row changes determinant by:
A) Increasing by that multiple
B) No change
C) Decreasing
D) Making it zero
Q9. If a 3×3 determinant’s value is 6, and the first row is multiplied by 4, the new determinant is:
A) 10
B) 24
C) 6
D) 18
Q10. If two columns are interchanged twice, the determinant becomes:
A) Same as original
B) Negative of original
C) Zero
D) Cannot be determined
✅ Answer Key Table
| Q.No | Answer |
|---|---|
| Q1 | C |
| Q2 | A |
| Q3 | A |
| Q4 | A |
| Q5 | B |
| Q6 | B |
| Q7 | B |
| Q8 | B |
| Q9 | B |
| Q10 | A |
🧠 Explanations of All Answers
- Q1 → C: Swapping two rows multiplies determinant by $-1$.
- Q2 → A: Identical rows ⇒ determinant is zero.
- Q3 → A: Multiplying one row by 5 multiplies determinant by 5.
- Q4 → A: Transpose doesn’t change determinant value.
- Q5 → B: Any row/column with all zeros ⇒ determinant = 0.
- Q6 → B: For triangular matrix, determinant = product of diagonal elements.
- Q7 → B: Multiplying all rows by $k$ ⇒ determinant × $k^n$.
- Q8 → B: Adding multiple of one row to another doesn’t change determinant.
- Q9 → B: Multiplying one row by 4 ⇒ determinant × 4 → $6 \times 4 = 24$.
- Q10 → A: Two swaps restore original determinant value.
🎯 Why This Practice Matters for ECET 2026
Properties of determinants save calculation time in exams. Instead of long expansions, using properties can help you answer in seconds. This topic often contributes 1–2 direct questions in ECET Mathematics, making it a high-return concept for quick marks.
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