Grade 11 Math I Chapter 5 – Matrices (Notes)

 Chapter 5 – Matrices (Notes)

5.1 Matrix Notation and Definitions

Definition

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

General form:

• Rows → horizontal
• Columns → vertical
• (a_{ij}) → element in row (i), column (j)

Order of Matrix

Order = Number of rows × Number of columns

Example:

Order =$2\times$$3$$\mathrm{<br>}$$\mathrm{<br>}$

$\mathrm{<br>}$

Types of Matrices

Row Matrix

$$[246]$$

(one row)

Column Matrix

(one column)

Square Matrix

Rows = Columns

Zero Matrix

All elements are zero.

Identity Matrix

Diagonal elements = 1, others = 0

5.2 Matrix Operations

Matrix Addition

Matrices can be added only when they have the same order.

Matrix Subtraction

Subtract corresponding elements.

$$A-B$$

Example:

calar Multiplication

Multiply every element by a constant.

5.3 Matrix Multiplication

Condition:

If matrix $A$ is $m\times n$ and matrix $B$ is $n\times p$, then multiplication is possible.

Example:

Important:

• $AB\neq BA$ generally
• Matrix multiplication is not commutative

5.4 Inverse of Square Matrix of Order 2

For matrix

Condition:

$$ad-bc\mathrm{\ne }0$$

If determinant = 0 → inverse does not exist.

Example:

Determinant:

$$(2)(4)-(1)(3)<br>$$$$=8-3$$$$=5$$

Inverse:

Quick Revision

✓ Matrix = arrangement of numbers in rows and columns
✓ Order = rows × columns
✓ Addition/subtraction → same order required
✓ Scalar multiplication → multiply every element
✓ Matrix multiplication → columns of first = rows of second
✓ $AB \neq BA$
✓ Determinant of $2\times2$ matrix = $ad-bc$
✓ Inverse exists only when determinant ≠ 0