Matrices and Determinants

Matrices and Determinants are powerful tools for solving systems of linear equations and transformations.

Overview

graph TD
    A[Matrices & Determinants] --> B[Matrices]
    A --> C[Determinants]
    B --> B1[Types]
    B --> B2[Operations]
    B --> B3[Inverse]
    C --> C1[Properties]
    C --> C2[Applications]

Matrices

A matrix is a rectangular array of numbers arranged in rows and columns.

Types of Matrices

Type	Description
Row Matrix	Only 1 row
Column Matrix	Only 1 column
Square Matrix	Rows = Columns
Diagonal Matrix	Non-zero only on diagonal
Scalar Matrix	Diagonal elements equal
Identity Matrix	Diagonal = 1, rest = 0
Null/Zero Matrix	All elements = 0
Symmetric	$A = A^T$
Skew-symmetric	$A = -A^T$

Matrix Operations

Addition: $(A + B)_{ij} = a_{ij} + b_{ij}$ (same order)

Scalar Multiplication: $(kA)_{ij} = k \cdot a_{ij}$

Matrix Multiplication:

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

Conditions: Columns of A = Rows of B

Properties

• $(AB)^T = B^T A^T$
• $(A^{-1})^T = (A^T)^{-1}$
• Every square matrix can be written as sum of symmetric and skew-symmetric matrices: $$A = \frac{A + A^T}{2} + \frac{A - A^T}{2}$$

Determinants

For a square matrix, determinant is a scalar value.

2×2 Determinant

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

3×3 Determinant

Expand along any row or column:

$$\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = a_1(b_2c_3 - b_3c_2) - b_1(a_2c_3 - a_3c_2) + c_1(a_2b_3 - a_3b_2)$$

Properties of Determinants

1. $|A^T| = |A|$
2. Interchanging rows/columns changes sign
3. Two identical rows/columns → $|A| = 0$
4. $|kA| = k^n|A|$ for n×n matrix
5. $|AB| = |A||B|$
6. Row operation: $R_i + kR_j$ doesn’t change det
7. Row operation: $kR_i$ multiplies det by k

Area of Triangle

Vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$:

$$\text{Area} = \frac{1}{2}\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}$$

Inverse of a Matrix

A matrix $A$ is invertible if $|A| \neq 0$

$$\boxed{A^{-1} = \frac{1}{|A|} \cdot adj(A)}$$

Adjoint Matrix

$adj(A)$ = transpose of cofactor matrix

Properties

• $AA^{-1} = A^{-1}A = I$
• $(AB)^{-1} = B^{-1}A^{-1}$
• $(A^{-1})^{-1} = A$
• $|A^{-1}| = \frac{1}{|A|}$
• $|adj(A)| = |A|^{n-1}$ for n×n matrix

Solving Linear Equations

For system $AX = B$:

Cramer’s Rule

$$x = \frac{D_x}{D}, \quad y = \frac{D_y}{D}, \quad z = \frac{D_z}{D}$$

where $D = |A|$, and $D_x, D_y, D_z$ are determinants with respective columns replaced by B.

Matrix Method

$$X = A^{-1}B$$

(if $|A| \neq 0$)

Consistency

Condition	System Type
$	A
$	A
$	A

Practice Problems

1. If $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, find $A^{-1}$.

2. Evaluate: $\begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{vmatrix}$

3. Solve using Cramer’s rule: $2x + y = 5$, $x - y = 1$

4. For what value of k does the system have no solution? $x + y + z = 1$, $x + 2y + 3z = 2$, $x + 2y + kz = 3$

Further Reading

• Complex Numbers
• Coordinate Geometry