Determinant of Matrix – Engineering Maths for GATE, ESE, AE/JE | Priyanka Ma’am

Determinant of a matrix is one of the most important topics in Linear Algebra under Engineering Mathematics. It is frequently asked in GATE, ESE, AE/JE exams and is used in finding inverse, solving equations, and checking matrix properties.

Basic Concept

The determinant is a scalar value associated with a square matrix.

Determinant of 2×2 Matrix

If A =

|A| = ad − bc

Determinant of 3×3 Matrix

If A =

|A| = a(ei − fh) − b(di − fg) + c(dh − eg)

Important Properties

• If any row or column is zero → Determinant = 0
• If two rows/columns are equal → Determinant = 0
• Interchanging two rows → sign changes
• If a row is multiplied by k → determinant multiplied by k
• |AB| = |A||B|
• |A^T| = |A|

Minors and Cofactors

Minor: Determinant obtained by deleting row and column

Cofactor: C_ij = (−1)^i+j × Minor

Solved Examples

Example 1

|A| = (1×4 − 2×3) = 4 − 6 = −2

Example 2

Answer = 3

Example 3

If two rows are equal, determinant?

Answer: 0

Example 4

If |A| = 5, find |2A| (n = order=2)

|2A| = 2ⁿ |A| (n = order=2) For 2×2 → 2² × 5 = 20

Shortcut Tricks

• Use row/column with maximum zeros
• Apply row operations to simplify
• Check zero row/duplicate row quickly
• Use property |kA| = kⁿ|A|

Questions

Q1. Determinant of identity matrix of order 3?

Answer: 1

Q2. If rows are proportional?

Answer: 0

Exam Tips

• Always simplify before expansion
• Use properties to save time
• Avoid full expansion when possible
• Practice PYQs regularly

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