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### Multiplication of Matrices

Two matrices are conformable for multiplication when the number of columns of the first matrix is equal to the number of rows of the second matrix.

For multiplication, multiply each element of a row of the first matrix by the corresponding element of the column of the second matrix and then add these products.

For multiplication, multiply each element of a row of the first matrix by the corresponding element of the column of the second matrix and then add these products.

Conformability for multiplication of matrices

Two matrices are conformable for multiplication when the number of columns of the first matrix equal the number of rows of the second matrix.

Show which of the following matrices are conformable for multiplication.

A=\left[\begin{array}{l}a \\ b\end{array}\right], B=\left[\begin{array}{ll}p & q\end{array}\right],

C=\left[\begin{array}{cc}1 & -1 \\ -2 & 1\end{array}\right] D=\left[\begin{array}{lll}p & r & s\end{array}\right]

1. AB

As number of Columns in matrix A=1

And number of Rows in matrix B=1

Thus AB is conformable for multiplication.

2. AC

As number of Columns in matrix A=1

And number of Rows in matrix C=2

Thus AC is not conformable for multiplication.

3. AD

As number of Columns in matrix A=1

And number of Rows in matrix D=1

Thus AD is conformable for multiplication.

4. BA

As number of Columns in matrix B=2

And number of Rows in matrix A=2

Thus BA is conformable for multiplication.

5. BC

As number of Columns in matrix B=2

And number of Rows in matrix C=2

Thus BC is conformable for multiplication.

6. BD

As number of Columns in matrix B=2

And number of Rows in matrix D=1

Thus BD is not conformable for multiplication.

7. CA

As number of Columns in matrix C=2

And number of Rows in matrix A=2

Thus CA is conformable for multiplication.

8. CB

As number of Columns in matrix C=2

And number of Rows in matrix B=1

Thus CB is not conformable for multiplication.

9. CD

As number of Columns in matrix C=2

And number of Rows in matrix D=1

Thus CD is not conformable for multiplication.

10. DA

As number of Columns in matrix D=3

And number of Rows in matrix A=2

Thus DA is not conformable for multiplication.

11. DB

As number of Columns in matrix D=3

And number of Rows in matrix B=1

Thus DB is not conformable for multiplication.

12. DC

As number of Columns in matrix D=3

And number of Rows in matrix C=2

Thus DC is not conformable for multiplication.

Two matrices are conformable for multiplication when the number of columns of the first matrix equal the number of rows of the second matrix.

Show which of the following matrices are conformable for multiplication.

A=\left[\begin{array}{l}a \\ b\end{array}\right], B=\left[\begin{array}{ll}p & q\end{array}\right],

C=\left[\begin{array}{cc}1 & -1 \\ -2 & 1\end{array}\right] D=\left[\begin{array}{lll}p & r & s\end{array}\right]

1. AB

As number of Columns in matrix A=1

And number of Rows in matrix B=1

Thus AB is conformable for multiplication.

2. AC

As number of Columns in matrix A=1

And number of Rows in matrix C=2

Thus AC is not conformable for multiplication.

3. AD

As number of Columns in matrix A=1

And number of Rows in matrix D=1

Thus AD is conformable for multiplication.

4. BA

As number of Columns in matrix B=2

And number of Rows in matrix A=2

Thus BA is conformable for multiplication.

5. BC

As number of Columns in matrix B=2

And number of Rows in matrix C=2

Thus BC is conformable for multiplication.

6. BD

As number of Columns in matrix B=2

And number of Rows in matrix D=1

Thus BD is not conformable for multiplication.

7. CA

As number of Columns in matrix C=2

And number of Rows in matrix A=2

Thus CA is conformable for multiplication.

8. CB

As number of Columns in matrix C=2

And number of Rows in matrix B=1

Thus CB is not conformable for multiplication.

9. CD

As number of Columns in matrix C=2

And number of Rows in matrix D=1

Thus CD is not conformable for multiplication.

10. DA

As number of Columns in matrix D=3

And number of Rows in matrix A=2

Thus DA is not conformable for multiplication.

11. DB

As number of Columns in matrix D=3

And number of Rows in matrix B=1

Thus DB is not conformable for multiplication.

12. DC

As number of Columns in matrix D=3

And number of Rows in matrix C=2

Thus DC is not conformable for multiplication.

Multiplication of Matrices

For multiplication, multiply each element of a row of the first matrix by the corresponding element of the column of the second matrix and then add these products.

OR

Multiply the first row of matrix A with each corresponding element of the first column of matrix B and then add these products.

A=\left[\begin{array}{cc}-1 & 0 \\ 2 & 1\end{array}\right], B=\left[\begin{array}{c}3 \\ -2\end{array}\right]

Is it possible to find AB?

Is it possible to find BA?

Find the possible products/ products.

Solution:

A=\left[\begin{array}{cc}-1 & 0 \\ 2 & 1\end{array}\right], B=\left[\begin{array}{c}3 \\ -2\end{array}\right]

AB

As number of Columns in matrix A=2

And number of Rows in matrix B=2

Thus A B is possible for multiplication.

BA

As number of Columns in matrix B=1

And number of Rows in matrix A=2

Thus BA is not possible for multiplication.

Now

A B=\left[\begin{array}{cc} -1 & 0 \\ 2 & 1 \end{array}\right]\left[\begin{array}{c} 3 \\ -2 \end{array}\right] \\

=\left[\begin{array}{c} (-1)(3)+(0)(-2) \\ (2)(3)+(1)(-2) \end{array}\right] \\

=\left[\begin{array}{c} -3+0 \\ 6+(-2) \end{array}\right] \\

=\left[\begin{array}{c} -3 \\ 6-2 \end{array}\right] \\

=\left[\begin{array}{c} -3 \\ 4 \end{array}\right]

For multiplication, multiply each element of a row of the first matrix by the corresponding element of the column of the second matrix and then add these products.

OR

Multiply the first row of matrix A with each corresponding element of the first column of matrix B and then add these products.

A=\left[\begin{array}{cc}-1 & 0 \\ 2 & 1\end{array}\right], B=\left[\begin{array}{c}3 \\ -2\end{array}\right]

Is it possible to find AB?

Is it possible to find BA?

Find the possible products/ products.

Solution:

A=\left[\begin{array}{cc}-1 & 0 \\ 2 & 1\end{array}\right], B=\left[\begin{array}{c}3 \\ -2\end{array}\right]

AB

As number of Columns in matrix A=2

And number of Rows in matrix B=2

Thus A B is possible for multiplication.

BA

As number of Columns in matrix B=1

And number of Rows in matrix A=2

Thus BA is not possible for multiplication.

Now

A B=\left[\begin{array}{cc} -1 & 0 \\ 2 & 1 \end{array}\right]\left[\begin{array}{c} 3 \\ -2 \end{array}\right] \\

=\left[\begin{array}{c} (-1)(3)+(0)(-2) \\ (2)(3)+(1)(-2) \end{array}\right] \\

=\left[\begin{array}{c} -3+0 \\ 6+(-2) \end{array}\right] \\

=\left[\begin{array}{c} -3 \\ 6-2 \end{array}\right] \\

=\left[\begin{array}{c} -3 \\ 4 \end{array}\right]

## Multiplication of Matrices MCQs with Explanation

**1. When number of columns of First matrix equal to number of rows of Second matrix is ________ for multiplication.**

O Conformable

O Not conformable

O Both a & b

O None of these

Answer: Conformable

Explanation:

Rules for Multiplication

**2. If A is m \times p \ \& \ B \ is \ p \times n then AB is ________ for multiplication.**

O Conformable

O Not conformable

O Both a & b

O None of these.

Answer: Conformable

Explanation:

This shows that A and B are conformable for multiplication because number of columns of first matrix equal to number of rows of second matrix. In short p=p

**3. If the order of A \ is \ m \times p \ \& \ B \ is \ p \times n then the order of AB is ________**

O m \times p

O p \times m

O m \times n

O p \times n

Answer: m \times n

Explanation:

This shows the order of product AB. Thus

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

**4. If A \ is \ p \times n \ \& \ B \ is \ m \times p , then AB is ________ for multiplication.**

O Conformable

O Not conformable

O Both a & b

O None of these.

Answer: Not Conformable

Explanation:

Here No. of rows of first matrix equal to number of columns of second matrix. Thus it is not conformable for multiplication.

**5. If A=\left[\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right], B=\left[\begin{array}{l}3 \\ 5\end{array}\right] , then AB is ________**

O Possible

O Not possible

O None of these

Answer: Possible

Explanation:

Here number of columns of A matrix equal to number of rows of B matrix. Thus, multiplication is possible.

**6. If A=\left[\begin{array}{l}3 \\ 2\end{array}\right], B=\left[\begin{array}{l}1 \\ 2\end{array}\right] then multiplication of matrices are ________**

O Possible

O Not possible

O Correct

O None

Answer: Not possible

Explanation:

Here number of columns of A matrix not equal to number of rows of B matrix. Thus, multiplication is not possible.

**7. Commutative law of multiplication of matrices may be ________**

O A B=B A

O A B \neq B A

O Both a & b

O None of these

Answer: Both a & b

Explanation:

In multiplication of matrices, Sometime Commutative law is possible but mostly not possible.

See Example 10 and 11 in KPK book Page No. 23

**8. A(BC)=(AB) C is called ________ law of multiplication.**

O Commutative

O Associative

O Distributive

O None of these

Answer: Associative

**9. Identity matrix is also known as ________ identity.**

O Additive

O Multiplicative

O All of these

O None of these

Answer: Multiplicative

Explanation:

When Identity matrix is multiplied to any matrix, the answer will be that matrix.

**10. (A+B)C=AC+AB is called ________ law of multiplication over addition.**

O Commutative

O Associative

O Distributive

O None of these

Answer: Distributive

**11. A I=I A= ________**

O A

O Null

O None

Answer: A

Explanation:

When Identity matrix “I” is multiplied to matrix “A”, the answer will be that matrix.

**12. B I= ________**

O A

O B

O I

O None

Answer: B

Explanation:

When Identity matrix “I” is multiplied to matrix “B”, the answer will be that matrix.

**13. Multiplicative identity of A=\left[\begin{array}{ll}1 & 3 \\ 4 & 5\end{array}\right] is ________**

O \left[\begin{array}{ll}1 & 4 \\ 3 & 5\end{array}\right]

O \left[\begin{array}{ll}1 & 4 \\ 3 & 5\end{array}\right]

O \left[\begin{array}{ll}-1 & -3 \\ -4 & -5\end{array}\right]

O \left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]

Answer: \left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]

Explanation:

The multiplicative identity for every matrix is:

\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]

**14. If A=\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6\end{array}\right] \ and \ I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \ then \ AI= ________**

O A

O I

O AI

O Not possible

Answer: Not possible

Explanation:

Here number of columns of A matrix not equal to number of rows of I matrix. Thus, multiplication is not possible.

**15. Transpose of \left[\begin{array}{ll}1 & 3 \\ 4 & 5\end{array}\right] is ________**

O \left[\begin{array}{ll}1 & 4 \\ 3 & 5\end{array}\right]

O \left[\begin{array}{ll}1 & 6 \\ 5 & 7\end{array}\right]

O \left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]

O All of these

Answer: \left[\begin{array}{ll}1 & 4 \\ 3 & 5\end{array}\right]

Explanation:

In transpoes, we interchanged the rows and columns.

**16. \left(A^t\right)^t= ________**

O A

O A^2

O All of these

Answer: A

Explanation:

If we take the transpose of a matrix and then again took the transpose, as a result the answer will be that matrix.

**17. \left(c^t\right)^t= ________**

O c^t

O -c

O I

O C

Answer: C

Explanation:

If we take the transpose of a matrix and then again took the transpose, as a result the answer will be that matrix.

**18. (A B)^t= ________**

O A^t B^t

O B^t A^t

O (B A)^t

O None of these

Answer: B^t A^t

Explanation:

The transpose of the product of the matrices is equal to the product of their transposes but in the reverse order.

**19. (B A)^t= ________**

O A^t B^t

O B^t A^t

O (A B)^t

O None of these

Answer: A^t B^t

Explanation:

The transpose of the product of the matrices is equal to the product of their transposes but in the reverse order.

**20. (A+B)^t= ________**

O A^t+B^t

O A^t-B^t

O (A B)^t

O A^t B^t

Answer: A^t+B^t

Explanation:

**21 . (A-B)^t= ________**

O A^t+B^t

O A^t-B^t

O (A B)^t

O A^t B^t

Answer: A^t-B^t

Explanation: