Matrices

ICSE Class 10 Mathematics • Chapter 07

1. Definition & Notation

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

Order: m × n (m rows, n columns). Written as $A_{m \times n}$

Elements: $a_{ij}$ denotes element in ith row and jth column.

2. Types of Matrices

Type Description Example
Row Matrix Only 1 row (1 × n) $[1 \quad 2 \quad 3]$
Column Matrix Only 1 column (m × 1) $\begin{bmatrix}1\\2\\3\end{bmatrix}$
Square Matrix Rows = Columns (n × n) $\begin{bmatrix}1&2\\3&4\end{bmatrix}$
Null/Zero Matrix All elements are 0 $\begin{bmatrix}0&0\\0&0\end{bmatrix}$
Diagonal Matrix Non-zero only on main diagonal $\begin{bmatrix}3&0\\0&5\end{bmatrix}$
Identity Matrix (I) 1s on diagonal, 0s elsewhere $\begin{bmatrix}1&0\\0&1\end{bmatrix}$

3. Equality of Matrices

Two matrices A and B are equal if:

  1. They have the same order
  2. Corresponding elements are equal: $a_{ij} = b_{ij}$ for all i, j

Example 1: If $\begin{bmatrix}x+3&y-1\\z&w+2\end{bmatrix} = \begin{bmatrix}5&4\\-1&7\end{bmatrix}$, find x, y, z, w.

$x + 3 = 5$ → $x = 2$

$y - 1 = 4$ → $y = 5$

$z = -1$

$w + 2 = 7$ → $w = 5$

4. Matrix Operations

A. Addition and Subtraction

Condition: Both matrices must have the same order.

Method: Add/subtract corresponding elements.

$(A + B)_{ij} = a_{ij} + b_{ij}$

B. Scalar Multiplication

Multiply every element by the scalar k:

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

Example 2: If $A = \begin{bmatrix}1&2\\3&4\end{bmatrix}$ and $B = \begin{bmatrix}5&6\\7&8\end{bmatrix}$, find $2A - B$.

$2A = \begin{bmatrix}2&4\\6&8\end{bmatrix}$

$2A - B = \begin{bmatrix}2-5&4-6\\6-7&8-8\end{bmatrix} = \begin{bmatrix}-3&-2\\-1&0\end{bmatrix}$

C. Matrix Multiplication

Condition: Columns of A = Rows of B

If A is $m \times n$ and B is $n \times p$, then AB is $m \times p$

Method: Row × Column, add products.

Steps for 2×2 Multiplication:

$\begin{bmatrix}a&b\\c&d\end{bmatrix} \times \begin{bmatrix}e&f\\g&h\end{bmatrix} = \begin{bmatrix}ae+bg&af+bh\\ce+dg&cf+dh\end{bmatrix}$

Example 3: Find $AB$ if $A = \begin{bmatrix}1&2\\3&4\end{bmatrix}$ and $B = \begin{bmatrix}2&0\\1&3\end{bmatrix}$

$AB = \begin{bmatrix}1(2)+2(1)&1(0)+2(3)\\3(2)+4(1)&3(0)+4(3)\end{bmatrix} = \begin{bmatrix}4&6\\10&12\end{bmatrix}$

5. Properties of Matrix Operations

Property Addition Multiplication
Commutative ✓ (A + B = B + A) ✗ (AB ≠ BA in general)
Associative ✓ ((A+B)+C = A+(B+C)) ✓ ((AB)C = A(BC))
Identity A + O = A AI = IA = A
Distributive A(B + C) = AB + AC

Important for Exams:

6. Transpose of Matrix

Transpose of A (written A' or A^T): Rows become columns, columns become rows.

If A is $m \times n$, then A' is $n \times m$

Quick Reference

Operation Condition Result Order
A + B Same order Same as A, B
kA Any matrix Same as A
A × B Cols(A) = Rows(B) Rows(A) × Cols(B)
A' Any matrix Cols(A) × Rows(A)

Exam Practice (PYQ Trends)

PYQ: 2023

BOARD If $A = \begin{bmatrix}2&-1\\0&3\end{bmatrix}$, $B = \begin{bmatrix}-1&2\\1&0\end{bmatrix}$, find $A^2 - 2B$.

PYQ: 2022

BOARD Find x and y if: $\begin{bmatrix}x&y\\4&8\end{bmatrix} + \begin{bmatrix}2&-5\\-3&6\end{bmatrix} = \begin{bmatrix}8&-2\\1&14\end{bmatrix}$