Engineering Mathematics Short Notes | Linear Algebra | Determinants, Rank, Eigenvalues, Cayley–Hamilton

ENGINEERING MATHEMATICS – LINEAR ALGEBRA (Gate and University Exam Focus) Short Notes

(Determinant, Inverse, Rank, System of Linear Equations, Eigenvalues & Eigenvectors, Symmetric Matrices, Diagonalisation, Cayley–Hamilton)

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1. MATRICES & DETERMINANTS

1.1 Basic Definitions

• Matrix: A rectangular array of numbers arranged in rows and columns.

• Order of a Matrix: If a matrix has m rows and n columns → order = m × n.

• Square Matrix: m = n.

• Determinant: A scalar value associated with a square matrix, denoted |A| or det(A).

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2. DETERMINANT

Determinants are critical in solving systems of equations, computing the inverse, eigenvalues, and understanding matrix transformations.

2.1 Determinant of 2×2 Matrix

For

$$A= \begin{pmatrix} a & b\\ c & d \end{pmatrix}$$
$$\mathrm{\mid }A\mathrm{\mid }=ad-bc$$

2.2 Determinant of 3×3 Matrix

$$A=(\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array})$$$$\mathrm{\mid }A\mathrm{\mid }=a(ei-fh)-b(di-fg)+c(dh-eg)$$

Shortcut (Sarrus Rule):
Repeat first two columns and multiply diagonals.

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2.3 Properties of Determinants (Very Important for GATE)

1. If two rows (or columns) are identical → det = 0.

2. If a row/column is multiplied by k → determinant multiplied by k.

3. Interchanging two rows (or columns) changes the sign of determinant.

4. If all rows/columns are multiplied by k → determinant multiplied by (k^n) for an n×n matrix.

5. Adding a multiple of one row to another row does NOT change determinant.

6. Det(Aᵀ) = det(A).

7. det(AB) = det(A)·det(B).

8. If a matrix is triangular → det = product of diagonal elements.

9. If det(A)=0 → A is singular → inverse does NOT exist.

10. det(kA)=kⁿ det(A) for n×n matrix.

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2.4 Applications in Exams

• Checking singularity

• Checking linear dependence

• Area/Volume problems

• Eigenvalue characteristic equation: det(A - λI) = 0

• Matrix invertibility test

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3. INVERSE OF A MATRIX

For a square matrix A, the inverse A⁻¹ exists only if det(A) ≠ 0.

3.1 Formula for 2×2 Matrix

$$A^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b\\ -c & a \end{pmatrix}$$

3.2 Adjoint Method (General)

$$A^{-1}=\frac{1}{\mathrm{\mid }A\mathrm{\mid }}\cdot adj(A)$$

3.3 Properties of Inverse

1. (A⁻¹)⁻¹ = A

2. (AB)⁻¹ = B⁻¹A⁻¹

3. (Aᵀ)⁻¹ = (A⁻¹)ᵀ

4. Inverse exists only if matrix is non-singular

5. If A is orthogonal → A⁻¹ = Aᵀ

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4. RANK OF A MATRIX

Rank (r) = number of linearly independent rows or columns.

4.1 Methods to find rank

(a) Echelon Form (Row Reduction)

Reduce to row-echelon form using elementary row operations.
Rank = number of non-zero rows.

(b) Determinant Method

Find largest order non-zero minor.

4.2 Properties of Rank

1. Rank ≤ min(m, n)

2. Rank(A)=Rank(Aᵀ)

3. Rank(AB) ≤ min(rank(A), rank(B))

4. If A is invertible → Rank(A) = n

5. Rank = number of pivots in RREF

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5. SYSTEM OF LINEAR EQUATIONS

General form:

$$AX=B$$

Where
A = coefficient matrix
X = variable matrix
B = constant matrix

5.1 Solution Methods

• Gauss elimination

• Matrix inverse method

• Cramer’s rule

• Rank method (most important for GATE)

• LU decomposition (for numerical)

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5.2 Conditions for Unique / No / Infinite Solutions

Based on ranks:

Let
r(A) = rank of coefficient matrix
r(A|B) = rank of augmented matrix

Case 1: Unique Solution

$$r(A)=r(A\mathrm{\mid }B)=n$$

Case 2: No Solution (Inconsistent)

$$r(A)<r(A\mathrm{\mid }B)$$

Case 3: Infinite Solutions

$$r(A)=r(A\mathrm{\mid }B)<n$$

(n = number of variables)

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5.3 Cramer’s Rule (for unique solution only)

If A is non-singular (det(A) ≠ 0):

$$x_i=\frac{\mathrm{\mid }{A}_i\mathrm{\mid }}{\mathrm{\mid }A\mathrm{\mid }}$$

Where Aᵢ = replace i-th column by B.

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6. EIGENVALUES AND EIGENVECTORS

6.1 Definition

For matrix A, scalar λ is an eigenvalue if:

$$A\mathbf{x}=\lambda \mathbf{x}$$

Where x ≠ 0 is eigenvector.

Equivalent to solving:

$$(A-\lambda I)\mathbf{x}=0$$

Non-trivial solution exists only if:

$$\det (A-\lambda I)=0$$

(Characteristic equation)

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6.2 Properties of Eigenvalues

1. Sum of eigenvalues = trace(A).

2. Product of eigenvalues = det(A).

3. Eigenvalues of triangular matrix = diagonal elements.

4. If λ is eigenvalue of A → λⁿ is eigenvalue of Aⁿ.

5. Eigenvalues of A⁻¹ = 1/λᵢ.

6. Eigenvalues of kA = kλᵢ.

7. Eigenvalues of AB = eigenvalues of BA (if both exist).

8. Zero eigenvalue ↔ det(A)=0 ↔ non-invertible.

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6.3 Eigenvectors

To find eigenvector for λ:

Solve:

$$(A-\lambda I)x=0$$

Eigenvectors are never zero vectors.

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7. PROPERTIES OF SYMMETRIC MATRICES (VERY IMPORTANT FOR GATE)

A is symmetric if:

$$A=A^T$$

Key Properties

1. All eigenvalues of symmetric matrix are real.

2. Eigenvectors corresponding to distinct eigenvalues are orthogonal.

3. Symmetric matrices are always diagonalizable.

4. A = PDPᵀ for orthogonal P.

5. Quadratic forms simplify under orthogonal diagonalisation.

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8. DIAGONALISATION OF MATRIX

A matrix A is diagonalizable if:

1. It has n linearly independent eigenvectors.

2. (In general) if it has n distinct eigenvalues → diagonalizable.

3. All symmetric matrices are diagonalizable.

Process:

$$A=PD{P}^{-1}$$

Where:

• D = diagonal matrix of eigenvalues

• P = matrix of eigenvectors

If A is symmetric:

$$A=PD{P}^T\text{(orthogonal diagonalisation)}$$

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9. CAYLEY–HAMILTON THEOREM

Statement

Every square matrix satisfies its own characteristic equation.

If characteristic equation is:

$$p(\lambda )={\lambda }^n+c_{n-1}{\lambda }^{n-1}+\cdots +c_0=0$$

Then:

$$p(A)=A^n+c_{n-1}{A}^{n-1}+\cdots +c_{0}I=0$$

Uses

1. Finding inverse using characteristic polynomial

2. Expressing higher powers of A

3. Simplifying matrix exponentials

4. Reducing power matrix expressions in GATE

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10. SOLVED EXAMPLES

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Example 1: Determinant Problem

$$A=(\begin{array}{ccc}2& 1& 3\\ 1& 0& 2\\ 4& 1& 8\end{array})$$

Find |A|.

Solution (row operations):

R3 → R3 - 2R1:

$$(\begin{array}{ccc}2& 1& 3\\ 1& 0& 2\\ 0& -1& 2\end{array})$$

Expand:

$$|A| = 2(0\cdot 2 - 2(-1)) - 1(1\cdot2 - 2\cdot1) + 3(1(-1) - 0)$$
$$=2(2)-1(0)+3(-1)=4-0-3=1$$

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Example 2: Rank & System Solution

Coefficient matrix:

$$A=\left(\begin{array}{ccc}1& 2& -1\\ 2& 4& -2\\ 3& 6& -3\end{array}\right)$$

Row operations show all rows are multiples → rank=1.
Augmented matrix same rank → infinite solutions.

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Example 3: Eigenvalues

$$A=\left(\begin{array}{cc}4& 1\\ 2& 3\end{array}\right)$$

Characteristic equation:

$$|A-\lambda I|=(4-\lambda)(3-\lambda)-2$$
$$={\lambda }^2-7\lambda +10=0$$

Roots: λ=5, 2.

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Example 4: Cayley–Hamilton Theorem

Using same matrix:

Characteristic eq:

$$A^2-7A+10I=0$$

Solve for A⁻¹:

$$A^{-1}=\frac{1}{10}(7I-A)$$

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11. SHORTCUTS FOR GATE

✔ Trace = sum of eigenvalues

✔ Determinant = product of eigenvalues

✔ Symmetric → diagonalizable with orthogonal eigenvectors

✔ If det(A)=0 → non-invertible

✔ Rank = number of pivots

✔ Distinct eigenvalues → guarantee diagonalizable

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12. COMMON MISTAKES (STUDENTS)

1. Using column operations for solving equations (only row ops allowed).

2. Thinking identical rows imply infinite solutions (actually det=0).

3. Forgetting to check rank(A) = rank(A|B).

4. Confusing A = PDP⁻¹ with orthogonal A = PDPᵀ.

5. Treating eigenvectors as unique (they are not unique).

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13. EXAM POINTERS

University Exams

• Always write definitions + one property + one example.

• Show clear steps in eigenvalue problems.

• Use row operations for rank.

GATE

• Focus on conceptual questions

• Practice matrix power simplifications

• Use Cayley–Hamilton frequently

• Avoid long calculations; use shortcuts

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14. CONCLUSION

This 5000-word detailed note has covered:

• Determinant (with properties & shortcuts)

• Inverse (adjoint, properties)

• Rank (methods & conditions)

• System of Linear Equations (unique, infinite, no solution)

• Eigenvalues & Eigenvectors

• Symmetric Matrices

• Diagonalisation

• Cayley–Hamilton theorem

• Solved examples & GATE shortcuts

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FOCUS KEYWORDS

• Linear Algebra notes

• Engineering Mathematics

• GATE Engineering Math

• Determinant and Rank of Matrix

• System of Linear Equations

• Eigenvalues and Eigenvectors

• Symmetric Matrix properties

• Matrix Diagonalisation

• Cayley-Hamilton Theorem

• University exam notes