Linear Algebra Notes (matrix & determinant)

April 04, 2025

Linear Algebra

Matrices & Determinants

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Table of Contents

• Matrices Fundamentals
• Matrix Operations
• Special Matrix Types
• Determinants
• Properties of Determinants
• Applications
• Practice Quiz

1. Matrices Fundamentals

A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are fundamental objects in linear algebra and have wide applications in mathematics, physics, engineering, and computer science.

$$ A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} $$

An m × n matrix has m rows and n columns. The element in the i-th row and j-th column is denoted as a_ij.

Matrix Notation

Matrices are typically denoted by uppercase letters (A, B, C, ...) and their elements by corresponding lowercase letters with subscripts.

Example: 2×3 Matrix

Consider matrix B:

$$ B = \begin{bmatrix} 1 & 4 & 0 \\ -2 & 3 & 5 \end{bmatrix} $$

Here, b₁₁ = 1, b₁₂ = 4, b₂₁ = -2, etc.

2. Matrix Operations

Matrix Addition

Two matrices of the same dimensions can be added by adding their corresponding elements.

$$ (A + B)_{ij} = A_{ij} + B_{ij} $$

Example

$$ \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 7 \\ 6 & 8 \end{bmatrix} = \begin{bmatrix} 1+5 & 3+7 \\ 2+6 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 10 \\ 8 & 12 \end{bmatrix} $$

Scalar Multiplication

A matrix can be multiplied by a scalar by multiplying each element by that scalar.

$$ (cA)_{ij} = c \cdot A_{ij} $$

Matrix Multiplication

For matrix multiplication AB, the number of columns in A must equal the number of rows in B.

$$ (AB)_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj} $$

Example

$$ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 1×5+2×7 & 1×6+2×8 \\ 3×5+4×7 & 3×6+4×8 \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix} $$

3. Special Matrix Types

Square Matrix

Number of rows equals number of columns (n × n)

$$ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} $$

Identity Matrix

Diagonal elements are 1, others are 0

$$ I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$

Diagonal Matrix

Non-diagonal elements are zero

$$ D = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix} $$

Symmetric Matrix

A = A^T (equal to its transpose)

$$ S = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{bmatrix} $$

4. Determinants

The determinant is a scalar value that can be computed from a square matrix and encodes important properties of the matrix.

2×2 Determinant

$$ \det \begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc $$

3×3 Determinant (Sarrus Rule)

$$ \det \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} = aei + bfg + cdh - ceg - bdi - afh $$

Example

$$ \det \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} = (1×5×9) + (2×6×7) + (3×4×8) - (3×5×7) - (2×4×9) - (1×6×8) $$ $$ = 45 + 84 + 96 - 105 - 72 - 48 = 0 $$

General n×n Determinant (Laplace Expansion)

$$ \det(A) = \sum_{j=1}^{n} (-1)^{i+j} a_{ij} \det(A_{ij}) $$

where A_ij is the submatrix obtained by removing the i-th row and j-th column.

5. Properties of Determinants

Key Properties

• Determinant of Identity: det(I) = 1
• Triangular Matrix: det = product of diagonal elements
• Product of Matrices: det(AB) = det(A)det(B)
• Transpose Property: det(A^T) = det(A)
• Scalar Multiplication: det(cA) = cⁿdet(A)
• Inverse Matrix: det(A^-1) = 1/det(A)

Example: Product Property

For matrices A and B:

$$ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} $$ $$ \det(A) = -2, \quad \det(B) = 1 $$ $$ AB = \begin{bmatrix} 2 & -1 \\ 4 & -3 \end{bmatrix}, \quad \det(AB) = -2 $$ $$ \det(A)\det(B) = -2 \times 1 = -2 $$

6. Applications

Solving Linear Systems (Cramer's Rule)

For a system Ax = b, if A is invertible:

$$ x_i = \frac{\det(A_i)}{\det(A)} $$

where A_i is A with the i-th column replaced by b.

Area/Volume Calculation

The absolute value of the determinant gives the scaling factor of the linear transformation described by the matrix.

$$ \text{Area} = \left| \det \begin{bmatrix} a & c \\ b & d \end{bmatrix} \right| $$

Eigenvalues

The determinant is used in finding eigenvalues:

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

7. Practice Quiz

Test your understanding with these GATE-style multiple choice questions:

1. What is the determinant of the following 2×2 matrix? $$ \begin{bmatrix} 3 & 4 \\ 1 & 2 \end{bmatrix} $$

2

5

10

2

Correct Answer: 2

The determinant is calculated as (3×2) - (4×1) = 6 - 4 = 2.

2. Which of the following is NOT a property of determinants?

det(AB) = det(A)det(B)

det(A+B) = det(A) + det(B)

det(A^T) = det(A)

det(cA) = cⁿdet(A) for n×n matrix A

Correct Answer: det(A+B) = det(A) + det(B)

Determinant of sum is not equal to sum of determinants. The determinant is multiplicative but not additive.

3. For a 3×3 matrix A with det(A) = 5, what is det(2A)?

10

40

5

2.5

Correct Answer: 40

For n×n matrix, det(kA) = kⁿdet(A). Here 2³×5 = 8×5 = 40.

4. What is the determinant of the identity matrix I₄?

0

1

4

Undefined

Correct Answer: 1

The determinant of any identity matrix is always 1 regardless of size.

0 Points