GATE CS - ENGINEERING MATHEMATICS:Linear Algebra

Linear Algebra for GATE CS

Linear Algebra usually contributes about 3–4 marks. Questions are short and formula-driven: eigenvalues of a small matrix, determinant shortcuts, rank / consistency of a linear system, and trace–determinant links. Treat this as easy marks if you can compute carefully under time pressure.

Recurring asks:

• Eigenvalues / eigenvectors (mostly $2 \times 2$)
• Determinant properties
• Rank and system consistency
• Trace = sum of eigenvalues; det = product

Matrix fundamentals

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 – $A^T = A^{-1}$

Tip:

If $\det(A) \neq 0$ → $A$ is invertible → rank is full → unique solution to $Ax = b$.

Fast 2×2 Multiplication Example

Let

• $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$
• $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$

Then

For $2 \times 2$ matrices in exam, use diagonal and cross products directly to save time.

Also remember: multiplication is not commutative, i.e. $AB \neq BA$ in general.

3. Determinants

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 $2 \times 2$

For

the determinant is:

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

3.2 Determinant Properties (GATE Goldmine)

Worth knowing cold:

• $\det(AB) = \det(A)\det(B)$
• $\det(A^T) = \det(A)$
• 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:

where:

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

4.1 Characteristic Equation

Eigenvalues are roots of:

For a $2 \times 2$ matrix, the characteristic polynomial can be written using trace and determinant:

where:

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

GATE Hack:

Avoid full expansion – use trace and determinant directly.

4.2 Exam-Type Example

Let

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

Characteristic equation:

Solving:

This is a 15‑second question in the exam.

4.3 Key Eigenvalue Properties

• Sum of eigenvalues $= \text{trace}(A)$
• 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:

Possibilities:

• Unique solution
• Infinite solutions
• No solution

Use rank to classify:

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

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 $\det(A) \neq 0$ for an $n \times n$ 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

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

• Eigenvalues = 2, 3

No computation needed → instant marks.

Q2. Rank / Consistency

System:

• 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 $3 \times 3$ matrix.

For $n \times n$:

Here, $n = 3, k = 3$:

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 $3 \times 3$ in the exam (usually a waste of time)

9. Rapid Revision Sheet (Night Before Exam)

Determinant

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

Eigenvalues

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

Rank

• Full rank $\Leftrightarrow \det(A) \neq 0$
• Zero/proportional rows reduce rank

System \(Ax = b\)

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

10. Practice Question Ideas (for Schoolabe)

You can turn these into quiz items:

1. Find eigenvalues of

$\begin{bmatrix} 4 & 1 \\ -2 & 3 \end{bmatrix}$

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

$\begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}$.

1. For what value of $k$ is the system

inconsistent?

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