Introduction to Matrices and Operations

Introduction to Matrices

A matrix is a rectangular arrangement of numbers, symbols or expressions arranged in rows and columns. प्रत्येक संख्या या expression को matrix का element कहते हैं।

For example:

\(A=\begin{pmatrix}2&5&7\\1&4&6\end{pmatrix}\)

This matrix has $2$ rows and $3$ columns.

Elements of a Matrix

In a matrix, the element in the $i$-th row and $j$-th column is denoted by $a_{ij}$.

For example, if

\(A=\begin{pmatrix}2&5&7\\1&4&6\end{pmatrix}\)

then $a_{11}=2$, $a_{12}=5$, $a_{13}=7$, $a_{21}=1$, $a_{22}=4$ and $a_{23}=6$.

Order of a Matrix

The order of a matrix is written as:

\(\text{Order}=\text{Number of rows}\times\text{Number of columns}\)

If a matrix has $m$ rows and $n$ columns, its order is $m\times n$.

For example,

\(A=\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}\)

has $2$ rows and $3$ columns. Therefore, the order of $A$ is $2\times3$.

General Form of a Matrix

A matrix of order $m\times n$ is generally represented as:

\(A=[a_{ij}]_{m\times n}\)

or

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

Here, $i=1,2,\ldots,m$ and $j=1,2,\ldots,n$.

Equality of Matrices

Two matrices $A$ and $B$ are equal if:

• They have the same order.
• Their corresponding elements are equal.

Thus,

$$A=B\iff a_{ij}=b_{ij}\quad\text{for all }i,j$$

For example, if

$$A=\begin{pmatrix}x&3\\5&y\end{pmatrix},\qquad B=\begin{pmatrix}2&3\\5&7\end{pmatrix}$$

then $A=B$ gives:

$$x=2,\qquad y=7$$

Types of Matrices

1. Row Matrix

A matrix having only one row is called a row matrix.

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

Its order is $1\times4$.

2. Column Matrix

A matrix having only one column is called a column matrix.

$$B=\begin{pmatrix}2\\4\\6\end{pmatrix}$$

Its order is $3\times1$.

3. Rectangular Matrix

A matrix in which the number of rows and columns are different is called a rectangular matrix.

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

Its order is $2\times3$.

4. Square Matrix

A matrix having the same number of rows and columns is called a square matrix.

$$A=\begin{pmatrix}1&2\\3&4\end{pmatrix}$$

Its order is $2\times2$.

5. Zero Matrix

A matrix in which every element is zero is called a zero matrix or null matrix.

$$O=\begin{pmatrix}0&0\\0&0\end{pmatrix}$$

6. Diagonal Matrix

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

$$A=\begin{pmatrix}2&0&0\\0&5&0\\0&0&7\end{pmatrix}$$

7. Scalar Matrix

A diagonal matrix in which all diagonal elements are equal is called a scalar matrix.

$$A=\begin{pmatrix}5&0&0\\0&5&0\\0&0&5\end{pmatrix}$$

8. Identity Matrix

A square matrix in which all principal diagonal elements are $1$ and all other elements are $0$ is called an identity matrix.

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

In general, the identity matrix of order $n$ is denoted by $I_n$.

Addition of Matrices

Two matrices can be added only when they have the same order. Their corresponding elements are added.

If

$$A=[a_{ij}]_{m\times n},\qquad B=[b_{ij}]_{m\times n}$$

then

$$A+B=[a_{ij}+b_{ij}]_{m\times n}$$

For example:

$$A=\begin{pmatrix}1&2\\3&4\end{pmatrix}, \qquad B=\begin{pmatrix}5&6\\7&8\end{pmatrix}$$ $$A+B= \begin{pmatrix} 1+5&2+6\\ 3+7&4+8 \end{pmatrix} = \begin{pmatrix} 6&8\\ 10&12 \end{pmatrix}$$

Subtraction of Matrices

Two matrices of the same order can be subtracted by subtracting their corresponding elements.

$$A-B=[a_{ij}-b_{ij}]$$

For example:

$$\begin{pmatrix}5&7\\9&11\end{pmatrix} - \begin{pmatrix}2&3\\4&5\end{pmatrix} = \begin{pmatrix}3&4\\5&6\end{pmatrix}$$

Scalar Multiplication

If $k$ is a scalar and $A=[a_{ij}]$ is a matrix, then every element of $A$ is multiplied by $k$.

$$kA=[ka_{ij}]$$

For example:

$$3\begin{pmatrix}1&2\\3&4\end{pmatrix} = \begin{pmatrix}3&6\\9&12\end{pmatrix}$$

Properties of Matrix Addition

For matrices of the same order:

Commutative Property

$$A+B=B+A$$

Associative Property

$$(A+B)+C=A+(B+C)$$

Additive Identity

$$A+O=O+A=A$$

Additive Inverse

$$A+(-A)=O$$

If

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

then

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

Multiplication of Matrices

Matrix multiplication is possible only when the number of columns of the first matrix is equal to the number of rows of the second matrix.

If $A$ is of order $m\times n$ and $B$ is of order $n\times p$, then $AB$ is defined and its order is:

$$AB=(m\times n)(n\times p)=m\times p$$

In multiplication, corresponding row elements of the first matrix are multiplied by corresponding column elements of the second matrix and then added.

Example of Matrix Multiplication

Let

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

Then

$$AB= \begin{pmatrix} 1(5)+2(7)&1(6)+2(8)\\ 3(5)+4(7)&3(6)+4(8) \end{pmatrix}$$ $$AB= \begin{pmatrix} 19&22\\ 43&50 \end{pmatrix}$$

Important Condition for Matrix Multiplication

If

$$A_{m\times n}\quad\text{and}\quad B_{p\times q}$$

then $AB$ is defined only when:

$$n=p$$

When multiplication is possible, the order of $AB$ is:

$$m\times q$$

Matrix Multiplication is Generally Not Commutative

For matrices, multiplication generally does not satisfy the commutative property.

$$AB\ne BA$$

Sometimes $AB$ may be defined while $BA$ is not defined. Even when both are defined, they may not be equal.

Associative Property of Matrix Multiplication

Matrix multiplication satisfies the associative property whenever the products are defined.

$$(AB)C=A(BC)$$

Distributive Properties

Matrix multiplication is distributive over addition.

Left Distributive Property

$$A(B+C)=AB+AC$$

Right Distributive Property

$$(A+B)C=AC+BC$$

Multiplication by Identity Matrix

If $A$ is a square matrix of appropriate order and $I$ is the identity matrix, then:

$$AI=IA=A$$

The identity matrix acts like $1$ in ordinary multiplication. Matrix multiplication by an identity matrix does not change the matrix.

Multiplication by Zero Matrix

Whenever the multiplication is defined:

$$AO=OA=O$$

Transpose of a Matrix

The transpose of a matrix is obtained by changing its rows into columns and its columns into rows.

The transpose of $A$ is denoted by $A^T$.

If

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

then

$$A^T=\begin{pmatrix}1&4\\2&5\\3&6\end{pmatrix}$$

If $A$ is of order $m\times n$, then $A^T$ is of order:

$$n\times m$$

Properties of Transpose

Transpose of Transpose

$$(A^T)^T=A$$

Transpose of a Sum

$$(A+B)^T=A^T+B^T$$

Transpose of a Scalar Multiple

$$(kA)^T=kA^T$$

Transpose of a Product

$$(AB)^T=B^T A^T$$

Symmetric Matrix

A square matrix $A$ is called a symmetric matrix if:

$$A^T=A$$

For example:

$$A=\begin{pmatrix}2&3\\3&5\end{pmatrix}$$

Since

$$A^T=\begin{pmatrix}2&3\\3&5\end{pmatrix}=A$$

therefore, $A$ is symmetric.

Skew-Symmetric Matrix

A square matrix $A$ is called a skew-symmetric matrix if:

$$A^T=-A$$

For example:

$$A=\begin{pmatrix}0&3\\-3&0\end{pmatrix}$$

Then

$$A^T=\begin{pmatrix}0&-3\\3&0\end{pmatrix}=-A$$

Therefore, $A$ is skew-symmetric.

Solved Example 1: Finding the Order

Find the order of:

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

There are $4$ rows and $3$ columns.

$$\therefore\ \text{Order of }A=4\times3$$

Solved Example 2: Equality of Matrices

If

$$\begin{pmatrix}x&2\\3&y\end{pmatrix} = \begin{pmatrix}5&2\\3&7\end{pmatrix}$$

compare corresponding elements:

$$x=5,\qquad y=7$$

Solved Example 3: Addition

Find $A+B$ if

$$A=\begin{pmatrix}2&4\\1&3\end{pmatrix}, \qquad B=\begin{pmatrix}5&1\\2&6\end{pmatrix}$$ $$A+B= \begin{pmatrix} 2+5&4+1\\ 1+2&3+6 \end{pmatrix} = \begin{pmatrix} 7&5\\ 3&9 \end{pmatrix}$$

Solved Example 4: Matrix Multiplication

Find $AB$ if

$$A=\begin{pmatrix}1&2\\2&1\end{pmatrix}, \qquad B=\begin{pmatrix}3&4\\5&6\end{pmatrix}$$ $$AB= \begin{pmatrix} 1(3)+2(5)&1(4)+2(6)\\ 2(3)+1(5)&2(4)+1(6) \end{pmatrix}$$ $$AB= \begin{pmatrix} 13&16\\ 11&14 \end{pmatrix}$$

How to Solve Matrix Questions

1. First identify the order of every matrix.
2. For addition or subtraction, check that the orders are the same.
3. For multiplication, check that the number of columns of the first matrix equals the number of rows of the second matrix.
4. For equality questions, compare corresponding elements.
5. For transpose, interchange rows and columns.
6. For symmetric matrices, verify $A^T=A$.
7. For skew-symmetric matrices, verify $A^T=-A$.

Important Examination Points

• Order of a matrix is written as rows $\times$ columns.
• Two matrices can be added or subtracted only when they have the same order.
• If $A$ is $m\times n$ and $B$ is $n\times p$, then $AB$ is $m\times p$.
• Matrix multiplication is generally not commutative: $AB\ne BA$.
• Matrix multiplication is associative: $(AB)C=A(BC)$.
• Matrix multiplication is distributive over addition.
• $AI=IA=A$ for an identity matrix of appropriate order.
• $(AB)^T=B^TA^T$.
• A symmetric matrix satisfies $A^T=A$.
• A skew-symmetric matrix satisfies $A^T=-A$.
• The diagonal elements of a skew-symmetric matrix are always zero.

Multiple Choice Questions

1. The order of the matrix

   $$\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}$$

   is:

   • (A) $2\times2$
   • (B) $2\times3$
   • (C) $3\times2$
   • (D) $3\times3$

2. Two matrices can be added only if they have:

   • (A) Equal determinants
   • (B) Same order
   • (C) Same number of elements
   • (D) Same diagonal elements

3. If $A$ is of order $2\times3$ and $B$ is of order $3\times4$, then the order of $AB$ is:

   • (A) $2\times4$
   • (B) $3\times3$
   • (C) $4\times2$
   • (D) $2\times3$

4. If $A$ is a square matrix, then $A^T=A$ represents:

   • (A) Zero matrix
   • (B) Diagonal matrix
   • (C) Symmetric matrix
   • (D) Skew-symmetric matrix

5. For a skew-symmetric matrix $A$:

   • (A) $A^T=A$
   • (B) $A^T=-A$
   • (C) $A^T=I$
   • (D) $A^T=O$

Fill in the Blanks

1. The order of a matrix is written as ______ $\times$ ______.
2. A matrix having one row is called a ______ matrix.
3. A square matrix satisfying $A^T=A$ is called a ______ matrix.
4. For a skew-symmetric matrix, $A^T=$ ______.
5. $(AB)^T=$ ______.

True or False

1. Two matrices of different orders can always be added.
2. Matrix multiplication is generally commutative.
3. $(AB)^T=B^TA^T$.
4. The diagonal elements of a skew-symmetric matrix are zero.
5. $AI=A$ when $I$ is the identity matrix of appropriate order.

Practice Questions

1. Find the order of the matrix $$A=\begin{pmatrix}2&4&6\\1&3&5\\7&8&9\end{pmatrix}.$$
2. If $$A=\begin{pmatrix}x&2\\3&y\end{pmatrix}$$ and $$B=\begin{pmatrix}5&2\\3&7\end{pmatrix},$$ find $x$ and $y$ when $A=B$.
3. Find $A+B$ if $$A=\begin{pmatrix}1&3\\2&4\end{pmatrix},\quad B=\begin{pmatrix}5&2\\6&1\end{pmatrix}.$$
4. Find $2A$ for $$A=\begin{pmatrix}2&-1\\3&4\end{pmatrix}.$$
5. Find $AB$ if $$A=\begin{pmatrix}1&2\\3&4\end{pmatrix},\quad B=\begin{pmatrix}2&1\\1&2\end{pmatrix}.$$
6. Find the transpose of $$A=\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}.$$
7. Check whether $$A=\begin{pmatrix}2&4\\4&7\end{pmatrix}$$ is symmetric.
8. Check whether $$A=\begin{pmatrix}0&5\\-5&0\end{pmatrix}$$ is skew-symmetric.
9. State whether the product $AB$ is defined when $A$ is of order $3\times2$ and $B$ is of order $3\times4$.
10. Verify the identity $(AB)^T=B^TA^T$ for two suitable matrices.

Quick Revision

Concept	Important Result
Order	$m\times n$
General Matrix	$A=[a_{ij}]_{m\times n}$
Addition	$A+B=[a_{ij}+b_{ij}]$
Scalar Multiplication	$kA=[ka_{ij}]$
Matrix Multiplication	$A_{m\times n}B_{n\times p}=AB_{m\times p}$
Identity	$AI=IA=A$
Transpose	$A^T$
Transpose of Product	$(AB)^T=B^TA^T$
Symmetric Matrix	$A^T=A$
Skew-Symmetric Matrix	$A^T=-A$