Introduction to matrices

Matrix – Class 9 | New Millennium Academy

New Millennium Academy · Birauta, Pokhara‑17 · Class 9

Matrix

Mathematics | Chapter: Matrix | Grade 9

1. Introduction to Matrix

Activity

Consider a classroom. Students sit in rows and columns. We represent each seat symbolically using numbers 1, 2, 3, 4, … (written as heads).

Observation

Numbers are arranged horizontally and vertically.
In Side 'A': students {1, 2}, {8, 7}, {9, 10} are sitting horizontally.
In Side 'B': students {3, 4}, {6, 5}, {11, 12} are sitting horizontally.
In Side 'A': students {1, 8, 9} and {2, 7, 10} are sitting vertically.
In Side 'B': students {3, 6, 11} and {4, 5, 12} are sitting vertically.
If we see horizontally, there are 6 horizontal rows.
If we see vertically, there are 4 vertical columns.

Conclusion

Those students sitting horizontally are sitting in rows.
Those students sitting vertically are sitting in columns.
The arrangement of numbers in rows and columns is actually a Matrix.

Real life examples of matrix: sitting arrangement in class, calendar on wall, grade sheet.

📌

Remember!

→ Horizontal arrangement = Rows
→ Vertical arrangement = Columns

Activity

Let us consider a set of numbers — marks obtained by 24 students in Opt-Maths out of 25 full marks:

      12, 9, 15, 24, 20, 18, 16, 17, 8, 11, 15, 12,

      10, 7, 6, 16, 18, 20, 11, 9, 5, 22, 23, 10

Question: How many different arrangements can be made?

(12, 2)12 rows × 2 numbers each

(2, 12)2 rows × 12 numbers each

(8, 3)8 rows × 3 numbers each

(3, 8)3 rows × 8 numbers each

(6, 4)6 rows × 4 numbers each

(4, 6)4 rows × 6 numbers each

Let us consider the (6, 4) arrangement:

12	9	15	24
20	18	16	17
8	11	15	12
10	7	6	16
18	20	11	9
5	22	23	10

Observation

Numbers are arranged in 6 horizontal rows and 4 vertical columns.
The numbers are enclosed inside brackets [ ].

Conclusion

Such an arrangement of numbers in rows and columns enclosed by brackets is called a Matrix.

Definition

The arrangement of numbers (or objects, alphabets, symbols, etc.) in rows and columns, enclosed by ( ) or [ ], is called a MATRIX.

Matrices are denoted by capital letters: $A, B, C, \ldots, X, Y, Z$.
Members of a matrix can be numbers, objects, alphabets, symbols, etc.

Example

$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} a & b & c & d \\ e & f & g & h \end{bmatrix}$$

2. Order of a Matrix

Activity

Let us consider these two matrices:

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

Observation

Matrix $A$ has 2 rows and 3 columns.
Matrix $B$ has 2 rows and 2 columns.

Conclusion

Rows and columns of a matrix form its order.

Definition

The order of any matrix is the number of rows and the number of columns, written as:

$$\text{Order} = \text{no. of rows} \times \text{no. of columns}$$

3. Entries (Elements) of a Matrix

Activity

Let us consider a $2 \times 3$ matrix:

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

Questions:

Q1.Where does 1 lie?

Q2.Where does 2 lie?

Q3.Where does 5 lie?

Observation (Answers)

A1. 1 lies on the 1st row and 1st column $\Rightarrow a_{11}$

A2. 2 lies on the 1st row and 2nd column $\Rightarrow a_{12}$

A3. 5 lies on the 2nd row and 2nd column $\Rightarrow a_{22}$

Conclusion

So, $a_{11},\; a_{12},\; a_{13},\; a_{21},\; a_{22},\; a_{23}$, … are called the entries of a matrix, which gives the location of each element in a matrix.

Definition

In general, a matrix $A$ is written as:

$$A = [a_{ij}]$$

where $i$ = row number and $j$ = column number.

Example – Construct entries for a 3×2 matrix

If the matrix is denoted by $P$, then its entries are:

$p_{11}$ $p_{12}$ $p_{21}$ $p_{22}$ $p_{31}$ $p_{32}$

In matrix form:

$$P = \begin{bmatrix} p_{11} & p_{12} \\ p_{21} & p_{22} \\ p_{31} & p_{32} \end{bmatrix}_{3 \times 2}$$

This matrix is called the Component Matrix.

A matrix expressed entirely in terms of its entry notations $a_{ij}$ is called a component matrix.

4. Types of Matrices

01 Row Matrix

Activity

Consider the following matrices and count their rows:

$$A = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & -2 & 0 & 7 \end{bmatrix}$$

Observation

Each of these matrices has only one row.

Conclusion

A matrix that has only one row is a Row Matrix.

Definition

A matrix having only one row is called a Row Matrix. Its order is $1 \times n$.

Example: $A = \begin{bmatrix} 3 & -1 & 5 \end{bmatrix}$ is a $1 \times 3$ row matrix.

02 Column Matrix

Activity

Consider the following matrices and count their columns:

$$A = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \quad B = \begin{bmatrix} 4 \\ -7 \end{bmatrix}$$

Observation

Each of these matrices has only one column.

Conclusion

A matrix that has only one column is a Column Matrix.

Definition

A matrix having only one column is called a Column Matrix. Its order is $m \times 1$.

Example: $B = \begin{bmatrix} 2 \\ 5 \\ -3 \end{bmatrix}$ is a $3 \times 1$ column matrix.

03 Null (Zero) Matrix

Activity

Consider the following matrices and look at their elements:

$$A = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 0 & 0 \end{bmatrix}$$

Observation

All elements (entries) in each matrix are zero.

Conclusion

A matrix whose all elements are zero is a Null or Zero Matrix.

Definition

A matrix in which all elements are zero is called a Null Matrix or Zero Matrix. It is denoted by $O$.

Example: $O = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

04 Rectangular Matrix

Activity

Consider the following matrices and compare their number of rows and columns:

$$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 7 & 8 \\ 9 & 10 \\ 11 & 12 \end{bmatrix}$$

Observation

Matrix $A$ has 2 rows and 3 columns → $2 \neq 3$.
Matrix $B$ has 3 rows and 2 columns → $3 \neq 2$.

In both cases, the number of rows is not equal to the number of columns.

Conclusion

A matrix where the number of rows $\neq$ number of columns is a Rectangular Matrix.

Definition

A matrix in which the number of rows is not equal to the number of columns $(m \neq n)$ is called a Rectangular Matrix.

05 Square Matrix

Activity

Consider the following matrices and compare their rows and columns:

$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$$

Observation

Matrix $A$: 2 rows, 2 columns → $2 = 2$.
Matrix $B$: 3 rows, 3 columns → $3 = 3$.

In both cases, the number of rows equals the number of columns.

Conclusion

A matrix where the number of rows equals the number of columns is a Square Matrix.

Definition

A matrix in which the number of rows is equal to the number of columns $(m = n)$ is called a Square Matrix of order $n$.

Example: $\begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$ is a square matrix of order 2.

📐 Key Concept: Diagonals of a Square Matrix

Before studying the remaining types of matrices, it is important to understand the two diagonals of a square matrix, as they are referred to frequently.

Principal Diagonal

The diagonal running from the top-left to the bottom-right of a square matrix is called the Principal Diagonal (also called main diagonal or leading diagonal). These are the elements where row number = column number, i.e., $a_{ij}$ where $i = j$.

Elements: $a_{11},\; a_{22},\; a_{33}$ (where $i = j$)

Secondary Diagonal

The diagonal running from the top-right to the bottom-left of a square matrix is called the Secondary Diagonal (also called anti-diagonal or counter-diagonal). These are the elements where $i + j = n + 1$ (for an $n \times n$ matrix).

Elements: $a_{13},\; a_{22},\; a_{31}$ (where $i + j = n + 1$)

Note: The element $a_{22}$ in the above 3×3 matrix lies on both the principal diagonal and the secondary diagonal. This happens only for the central element in odd-ordered square matrices.

06 Diagonal Matrix

Activity

Consider the following square matrices and observe their non-diagonal elements:

$$A = \begin{bmatrix} 3 & 0 \\ 0 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 7 \end{bmatrix}$$

Observation

All elements except the diagonal elements (i.e., elements where $i = j$) are zero.

Conclusion

A square matrix in which all non-diagonal elements are zero is a Diagonal Matrix.

Definition

A square matrix in which all the elements except the principal diagonal elements are zero is called a Diagonal Matrix.

That is, $a_{ij} = 0$ for all $i \neq j$.

07 Scalar Matrix

Activity

Consider the following diagonal matrices and observe the diagonal elements:

$$A = \begin{bmatrix} 4 & 0 \\ 0 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix}$$

Observation

All diagonal elements are equal to each other (and non-diagonal elements are zero).

Conclusion

A diagonal matrix in which all diagonal elements are equal (same constant) is a Scalar Matrix.

Definition

A diagonal matrix in which all the diagonal elements are equal $(a_{ii} = k \text{ for all } i)$ is called a Scalar Matrix.

08 Identity Matrix (Unit Matrix)

Activity

Consider the following scalar matrices and observe the value of the diagonal elements:

$$I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

Observation

All diagonal elements are 1 and all non-diagonal elements are 0.

Conclusion

A scalar matrix where every diagonal element equals 1 is an Identity (Unit) Matrix.

Definition

A square matrix in which all diagonal elements are 1 and all other elements are 0 is called an Identity Matrix or Unit Matrix, denoted by $I$.

That is, $a_{ij} = 1$ if $i = j$, and $a_{ij} = 0$ if $i \neq j$.

09 Triangular Matrix (Upper and Lower)

Activity

Consider the following square matrices and observe the position of zero and non-zero elements:

$$U = \begin{bmatrix} 2 & 3 & 5 \\ 0 & 4 & 6 \\ 0 & 0 & 1 \end{bmatrix}, \quad L = \begin{bmatrix} 2 & 0 & 0 \\ 3 & 4 & 0 \\ 5 & 6 & 1 \end{bmatrix}$$

Observation

In matrix $U$, all elements below the principal diagonal are zero.
In matrix $L$, all elements above the principal diagonal are zero.

Conclusion

A square matrix with all zeros below the diagonal is an Upper Triangular Matrix.
A square matrix with all zeros above the diagonal is a Lower Triangular Matrix.

Definition

A square matrix is called an Upper Triangular Matrix if all elements below the principal diagonal are zero ($a_{ij} = 0$ for $i > j$).

A square matrix is called a Lower Triangular Matrix if all elements above the principal diagonal are zero ($a_{ij} = 0$ for $i < j$).