Linear Algebra – GATE CS Premium Module

A complete, exam-focused, concept-driven guide designed to help you score full marks in this chapter. Every section below is written with GATE CS question patterns in mind.

1. Why Linear Algebra Matters in GATE

Linear Algebra consistently contributes 3–4 marks every year in GATE CS.

The best part?

• Questions are formula-based, predictable, and high-accuracy
• Once concepts are clear, this becomes one of the easiest scoring topics

You will mostly see questions from:

• Eigenvalues and eigenvectors (mostly (2 imes 2) matrices)
• Determinant shortcuts
• Rank and system consistency
• Diagonalization / matrix properties
• Trace–determinant relationships

Idea: Master this chapter → secure guaranteed marks.

2. Matrix Fundamentals (with GATE Intuition)

A matrix is best seen as a linear transformation, not just a grid of numbers.

Think of a matrix (A) as an operation that:

• Rotates
• Scales
• Shears

a vector in some direction.

Eigenvalues tell how much it scales vectors in special directions (eigenvectors).

2.1 Matrix Operations (as required for GATE)

You must be fluent in:

• Addition / subtraction – only for same-sized matrices
• Multiplication – row of (A) × column of (B)
• Transpose – flips rows and columns
• Inverse – only for square, non-singular matrices
• Orthogonal matrices – (A^T = A^{-1})

GATE Trick:

If (det(A)

eq 0) → (A) is invertible → rank is full → unique solution to (Ax = b).

Fast 2×2 Multiplication Example

Let

• (A = egin{bmatrix} 1 & 2 \ 3 & 4 end{bmatrix})
• (B = egin{bmatrix} 5 & 6 \ 7 & 8 end{bmatrix})

Then

[

A B =

egin{bmatrix}

1cdot 5 + 2cdot 7 & 1cdot 6 + 2cdot 8 \

3cdot 5 + 4cdot 7 & 3cdot 6 + 4cdot 8

end{bmatrix}

=

egin{bmatrix}

19 & 22 \

43 & 50

end{bmatrix}

]

For (2 imes 2) matrices in exam, use diagonal and cross products directly to save time.

Also remember: multiplication is not commutative, i.e. (AB

eq BA) in general.

3. Determinants (Pure Scoring Zone)

The determinant is a single number that tells you:

• Whether a matrix is invertible
• How area/volume scales under the transformation
• Whether a system has unique / infinite / no solution

3.1 Determinant of (2 imes 2)

For

[

A =

egin{bmatrix}

a & b \\

c & d

end{bmatrix}

]

the determinant is:

[

|A| = ad - bc

]

GATE will almost surely ask at least one direct question using this.

3.2 Determinant Properties (GATE Goldmine)

Memorise these:

• (det(AB) = det(A)det(B))
• (det(A^T) = det(A))
• Swapping two rows → changes sign of determinant
• Multiplying a row by (k) → determinant is multiplied by (k)
• Two identical (or proportional) rows → (det(A) = 0)
• If a row is a linear combination of others → (det(A) = 0)

GATE Hack:

If (det(A) = 0) →

• Matrix is singular
• Rank is not full
• System (Ax = b) has infinite or no solutions

This lets you kill options very quickly.

4. Eigenvalues & Eigenvectors (Most Tested Area)

Eigenvalues tell how a matrix scales certain special directions (eigenvectors).

Definition:

[

Av = lambda v

]

where:

• (lambda) = eigenvalue
• (v) = eigenvector (non-zero)

4.1 Characteristic Equation

Eigenvalues are roots of:

[

|A - lambda I| = 0

]

For a (2 imes 2) matrix, the characteristic polynomial can be written using trace and determinant:

[

lambda^2 - ( ext{trace}),lambda + det(A) = 0

]

where:

• ( ext{trace} = a + d) (sum of diagonal entries)
• (det(A) = ad - bc)

GATE Hack:

Avoid full expansion – use trace and determinant directly.

4.2 Exam-Type Example

Let

[

A =

egin{bmatrix}

3 & 2 \\

1 & 4

end{bmatrix}

]

• ( ext{trace} = 3 + 4 = 7)
• (det(A) = 3cdot 4 - 2cdot 1 = 10)

Characteristic equation:

[

lambda^2 - 7lambda + 10 = 0

]

Solving:

[

(lambda - 5)(lambda - 2) = 0 Rightarrow lambda = 5, 2

]

This is a 15‑second question in the exam.

4.3 Key Eigenvalue Properties

• Sum of eigenvalues (= ext{trace}(A))
• Product of eigenvalues (= det(A))
• Eigenvalues of (A^k) are (lambda^k)
• Eigenvalues of (A^{-1}) are (1/lambda) (for non-zero (lambda))
• If (A) is triangular (upper/lower), eigenvalues are just the diagonal entries
• If (A) is orthogonal, then (|lambda| = 1)

These shortcuts solve most eigenvalue questions without heavy calculation.

5. System of Linear Equations (Using Rank)

Consider a system:

[

Ax = b

]

Possibilities:

• Unique solution
• Infinite solutions
• No solution

Use rank to classify:

• Unique solution: ( ext{rank}(A) = ext{rank}([A|b]) = n)
• Infinite solutions: ( ext{rank}(A) = ext{rank}([A|b]) < n)
• No solution: ( ext{rank}(A) < ext{rank}([A|b]))

GATE frequently tests this classification.

6. Rank of a Matrix

Rank = number of linearly independent rows or columns.

6.1 Quick Rank Clues (GATE Pattern)

• A row/column of all zeros → reduces rank
• Two proportional rows → dependence → reduces rank
• If (det(A)

eq 0) for an (n imes n) matrix → full rank (= n)

• Elementary row operations do not change rank

These ideas are enough for most GATE rank questions.

7. GATE PYQ‑Style Questions (Solved)

Q1. Eigenvalues of a Triangular Matrix

Let

[

A =

egin{bmatrix}

2 & 1 \\

0 & 3

end{bmatrix}

]

Since (A) is upper triangular, eigenvalues are just the diagonal entries:

• Eigenvalues = 2, 3

No computation needed → instant marks.

Q2. Rank / Consistency

System:

[

egin{cases}

x + y + z = 1 \\

2x + 2y + 2z = 2 \\

x + y + z = 3

end{cases}

]

• Row 2 is 2 × Row 1 (proportional)
• Row 1 and Row 3 contradict (same left side, different right side)

⇒ System is inconsistent → no solution.

Q3. Determinant Scaling

If (det(A) = 5), find (det(3A)) for a (3 imes 3) matrix.

For (n imes n):

[

det(kA) = k^n det(A)

]

Here, (n = 3, k = 3):

[

det(3A) = 3^3 cdot 5 = 27 cdot 5 = 135

]

8. Common Mistakes (Mark Destroyers)

Avoid these:

• Assuming (AB = BA) for arbitrary matrices
• Forgetting sign change when swapping rows in determinants
• Expanding big determinants instead of using properties
• Ignoring the fact that triangular matrices have eigenvalues on the diagonal
• Thinking rank changes under elementary row operations (it does not)
• Using full inverse formulas for (3 imes 3) in the exam (usually a waste of time)

9. Rapid Revision Sheet (Night Before Exam)

Determinant

• (2 imes 2): (|A| = ad - bc)
• (det(AB) = det(A)det(B))
• (det(A^T) = det(A))

Eigenvalues

• From trace/determinant: (lambda^2 - ( ext{trace})lambda + det(A) = 0)
• Triangular matrix → eigenvalues = diagonal entries
• Eigenvalues of (A^k): (lambda^k)
• Eigenvalues of (A^{-1}): (1/lambda)

Rank

• Full rank ( Leftrightarrow det(A)

eq 0)

• Zero/proportional rows reduce rank

System (Ax = b)

• Unique: ( ext{rank}(A) = ext{rank}([A|b]) = n)
• Infinite: ( ext{rank}(A) = ext{rank}([A|b]) < n)
• None: ( ext{rank}(A) < ext{rank}([A|b]))

10. Practice Question Ideas (for Schoolabe)

You can turn these into quiz items:

1. Find eigenvalues of

(egin{bmatrix} 4 & 1 \\ -2 & 3 end{bmatrix})

1. If (det(A) = -4) for a (3 imes 3) matrix, find (det(2A)).
2. Determine rank of

(egin{bmatrix} 1 & 2 \\ 2 & 4 end{bmatrix}).

1. For what value of (k) is the system

[

x + y = 2,quad 2x + 2y = k

]

inconsistent?

1. If eigenvalues of (A) are 1 and (-2), find eigenvalues of (A^3).