Matrix Multiplication

Matrix Multiplication

Matrix multiplication is an operation in which the elements of one matrix are multiplied with the elements of another matrix according to the row-column rule. इसमें पहले matrix की row और दूसरे matrix की column को multiply करके उनके products का sum लिया जाता है।

Condition for Matrix Multiplication

Let $A$ be a matrix of order $m\times n$ and $B$ be a matrix of order $n\times p$.

\(A_{m\times n}B_{n\times p}=AB_{m\times p}\)

Thus, the number of columns of the first matrix must be equal to the number of rows of the second matrix.

Order of the Product Matrix

If

\(A_{m\times n}\quad\text{and}\quad B_{n\times p}\)

then the product $AB$ has order:

\(AB_{m\times p}\)

For example, if $A$ is $2\times3$ and $B$ is $3\times4$, then:

\(AB=(2\times3)(3\times4)=2\times4\)

Therefore, the product matrix will have $2$ rows and $4$ columns.

How Matrix Multiplication Works

Consider two matrices:

$$$A=\begin{pmatrix}a&b\c&d\end{pmatrix}, \qquad B=\begin{pmatrix}p&q\\r&s\end{pmatrix}$$

Then:

$$AB= \begin{pmatrix} ap+br&aq+bs\\ cp+dr&cq+ds \end{pmatrix}$$

Notice that each element of $AB$ is obtained by multiplying a row of $A$ by a column of $B$.

Step-by-Step Example

Let

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

To find $AB$, multiply the first row of $A$ by the first column of $B$:

$$1(5)+2(7)=5+14=19$$

First row of $A$ × second column of $B$:

$$1(6)+2(8)=6+16=22$$

Second row of $A$ × first column of $B$:

$$3(5)+4(7)=15+28=43$$

Second row of $A$ × second column of $B$:

$$3(6)+4(8)=18+32=50$$

Therefore:

$$AB=\begin{pmatrix}19&22\\43&50\end{pmatrix}$$

Matrix Multiplication of a $2\times3$ and $3\times2$ Matrix

Let

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

Here, $A$ is of order $2\times3$ and $B$ is of order $3\times2$. Therefore, $AB$ is defined and its order is $2\times2$.

$$AB= \begin{pmatrix} 1(7)+2(9)+3(11)&1(8)+2(10)+3(12)\\ 4(7)+5(9)+6(11)&4(8)+5(10)+6(12) \end{pmatrix}$$ $$AB= \begin{pmatrix} 58&64\\ 139&154 \end{pmatrix}$$

Matrix Multiplication Using Row and Column

If

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

then the element in the $i$-th row and $j$-th column of $AB$ is:

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

This formula expresses the complete row-column rule mathematically.

Example Using the General Formula

Let

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

The element $(AB)_{11}$ is:

$$(AB)_{11}=1(1)+2(3)+3(5)=22$$

The element $(AB)_{12}$ is:

$$(AB)_{12}=1(2)+2(4)+3(6)=28$$

The element $(AB)_{21}$ is:

$$(AB)_{21}=4(1)+5(3)+6(5)=49$$

The element $(AB)_{22}$ is:

$$(AB)_{22}=4(2)+5(4)+6(6)=64$$

Hence:

$$AB=\begin{pmatrix}22&28\\49&64\end{pmatrix}$$

Matrix Multiplication is Not Commutative

For ordinary numbers, multiplication is commutative:

$$ab=ba$$

But matrix multiplication is generally not commutative:

$$AB\ne BA$$

This means that even if both $AB$ and $BA$ are defined, they may not be equal.

Example of Non-Commutativity

Let

$$A=\begin{pmatrix}1&2\\3&4\end{pmatrix}, \qquad B=\begin{pmatrix}2&0\\1&2\end{pmatrix}$$

Then:

$$AB= \begin{pmatrix} 1(2)+2(1)&1(0)+2(2)\\ 3(2)+4(1)&3(0)+4(2) \end{pmatrix} = \begin{pmatrix} 4&4\\ 10&8 \end{pmatrix}$$

But:

$$BA= \begin{pmatrix} 2(1)+0(3)&2(2)+0(4)\\ 1(1)+2(3)&1(2)+2(4) \end{pmatrix} = \begin{pmatrix} 2&4\\ 7&10 \end{pmatrix}$$

Therefore:

$$AB\ne BA$$

Associative Property

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

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

This means that the grouping of matrices can be changed, but the order of the matrices cannot be changed.

Distributive Properties

Matrix multiplication is distributive over matrix addition.

Left Distributive Property

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

Right Distributive Property

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

Multiplication by Identity Matrix

Let $I$ be the identity matrix of appropriate order. Then:

$$AI=IA=A$$

For example:

$$A=\begin{pmatrix}2&3\\4&5\end{pmatrix}, \qquad I_2=\begin{pmatrix}1&0\\0&1\end{pmatrix}$$

Then:

$$AI_2= \begin{pmatrix}2&3\\4&5\end{pmatrix} \begin{pmatrix}1&0\\0&1\end{pmatrix} = \begin{pmatrix}2&3\\4&5\end{pmatrix} =A$$

Multiplication by Zero Matrix

Whenever the multiplication is defined:

$$AO=OA=O$$

For example:

$$\begin{pmatrix}1&2\\3&4\end{pmatrix} \begin{pmatrix}0&0\\0&0\end{pmatrix} = \begin{pmatrix}0&0\\0&0\end{pmatrix}$$

When Matrix Product is Not Defined

Suppose

$$A_{2\times3} \qquad\text{and}\qquad B_{2\times2}$$

For $AB$, the number of columns of $A$ is $3$, while the number of rows of $B$ is $2$.

$$3\ne2$$

Therefore, $AB$ is not defined.

Important Difference Between $AB$ and $BA$

Point	$AB$	$BA$
Order condition	Columns of $A$ = Rows of $B$	Columns of $B$ = Rows of $A$
Product order	If defined, depends on orders of $A$ and $B$	If defined, may have a different order
Equality	$AB$ is generally not equal to $BA$

Important Properties of Matrix Multiplication

Property	Formula
Associative	$(AB)C=A(BC)$
Left Distributive	$A(B+C)=AB+AC$
Right Distributive	$(A+B)C=AC+BC$
Identity	$AI=IA=A$
Zero Matrix	$AO=OA=O$
Commutative	Generally, $AB\ne BA$

How to Solve Matrix Multiplication Questions

1. Write the order of both matrices.
2. Check whether the number of columns of the first matrix equals the number of rows of the second matrix.
3. Determine the order of the product matrix.
4. Multiply each row of the first matrix with each column of the second matrix.
5. Add the products to obtain each element.
6. Arrange the calculated elements in their correct positions.

Common Mistakes

• Do not multiply corresponding elements directly.
• Do not forget to check the multiplication condition.
• Do not assume $AB=BA$.
• Do not change the order of matrices while applying the associative property.
• Remember that $(AB)^T=B^TA^T$, not $A^TB^T$.

Multiple Choice Questions

1. 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) $2\times3$

2. The product $AB$ is defined when:

   • (A) Rows of $A$ = Rows of $B$
   • (B) Columns of $A$ = Columns of $B$
   • (C) Columns of $A$ = Rows of $B$
   • (D) Rows of $A$ = Columns of $A$

3. Which property is represented by $(AB)C=A(BC)$?

   • (A) Commutative
   • (B) Associative
   • (C) Distributive
   • (D) Inverse

4. In general, which statement is correct?

   • (A) $AB=BA$
   • (B) $AB\ne BA$
   • (C) $AB=I$
   • (D) $AB=O$

5. If $I$ is the identity matrix of appropriate order, then:

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

Fill in the Blanks

1. If $A$ is $m\times n$ and $B$ is $n\times p$, then $AB$ is of order ______.
2. Matrix multiplication is generally not ______.
3. $(AB)C=A(BC)$ represents the ______ property.
4. $A(B+C)=$ ______.
5. $(AB)^T=$ ______.

True or False

1. Matrix multiplication is always commutative.
2. If $A$ is $2\times3$ and $B$ is $3\times4$, then $AB$ is defined.
3. If $A$ is $2\times3$ and $B$ is $2\times4$, then $AB$ is defined.
4. $(AB)C=A(BC)$ whenever the products are defined.
5. $AI=A$ for an identity matrix $I$ of appropriate order.

Practice Questions

1. Find $AB$ if $$A=\begin{pmatrix}1&2\\3&4\end{pmatrix},\quad B=\begin{pmatrix}5&6\\7&8\end{pmatrix}.$$
2. Find $BA$ for the matrices in Question 1 and verify that generally $AB\ne BA$.
3. Find $AB$ if $$A=\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix},\quad B=\begin{pmatrix}1&2\\3&4\\5&6\end{pmatrix}.$$
4. Determine whether the product $AB$ is defined if $A$ is of order $3\times2$ and $B$ is of order $3\times4$.
5. If $A$ is of order $4\times3$ and $B$ is of order $3\times5$, find the order of $AB$.
6. Verify $A(B+C)=AB+AC$ for suitable matrices.
7. Verify $(AB)^T=B^TA^T$ for suitable matrices.
8. Find $A^2$ if $$A=\begin{pmatrix}2&1\\1&2\end{pmatrix}.$$

Quick Revision

Concept	Result
Multiplication Condition	$\text{Columns of }A=\text{Rows of }B$
Order of Product	$(m\times n)(n\times p)=m\times p$
General Element	$(AB)_{ij}=\sum_{k=1}^{n}a_{ik}b_{kj}$
Associative Property	$(AB)C=A(BC)$
Distributive Property	$A(B+C)=AB+AC$
Identity	$AI=IA=A$
Zero Matrix	$AO=OA=O$
Commutative Property	Generally, $AB\ne BA$