# Worksheet: Matrix Multiplication

In this worksheet, we will practice identifying the conditions for matrix multiplication and evaluating the product of two matrices if possible.

Q1:

Given that find .

• A
• B
• C
• D

Q2:

Given that find if possible.

• A
• BIt is not possible.
• C
• D
• E

Q3:

Given that , find .

• A
• B
• C
• D

Q4:

Given that find if possible.

• A
• B
• C
• Dit is not possible
• E

Q5:

Consider the matrices Find , if possible.

• A
• B
• C
• D
• EIt is not possible.

Q6:

Consider the matrices

Find if possible.

• A
• B
• C
• D

Q7:

Consider the matrices Find and if possible.

• A,
• BIt is not possible.
• C,
• D,
• E,

Q8:

Consider the matrices

Find .

• A
• B
• C
• D
• E

Find .

• A
• B
• C
• D
• E

Q9:

Suppose

Find the product .

• A
• B
• C
• D
• E

Find the product .

• A
• B
• C
• D
• E

Q10:

Evaluate the matrix product

• A
• B
• C
• D
• E

Q11:

Consider the matrix product

What can you conclude about it?

• AFor a given matrix , there cannot be any matrix except the identity matrix for which .
• BFor a given matrix , there can be a matrix that is not the identity matrix for which .
• CFor a given matrix , there can be a matrix that is not the identity matrix for which .
• DFor a given matrix , there can be a matrix that is not the identity matrix for which .

Is it possible to find a matrix with the above property for every matrix ?

• Ano
• Byes

Q12:

Consider the matrices

Find and .

• A,
• B,
• C,
• D,

Q13:

Let and . Find and .

• A,
• B,
• C, 1
• D, 1
• E, 1

Q14:

Consider the matrices Find if possible.

• A
• B
• C
• D

Q15:

Given that and is the unit matrix of the same order, find for which .

Q16:

Given that find if possible.

• A
• B
• C
• D

Q17:

Given that and , find if possible.

• A
• B
• C
• D

Q18:

Consider the matrices Find , if possible.

• A
• B
• C
• D

Q19:

Given that determine if possible.

• A
• Bundefined
• C
• D

Q20:

Is it possible to have a matrix and a matrix such that If so, give an example.

• Ayes, ;
• Bno
• Cyes, ;

Q21:

Suppose the matrix product makes sense. We also know that has 2 rows, has 3 columns, and has 4 entries. Is it possible to determine the possible sizes of these matrices? If so, what are the possible sizes of , , and ?

• Ayes, , , ; , , ; , ,
• Byes, , , ; , , ; , ,
• Cyes, , , ; , , ; , ,
• Dno
• Eyes, , , ; , , ; , ,

Q22:

Find the matrices and such that, for any matrix , and . Explain why and are not the same.

• A, , and have different dimensions.
• B, , and have different dimensions.
• C, , and have different dimensions.
• D, , and have different dimensions.
• E, , and have different dimensions.

Q23:

Given that is a matrix of order and is a matrix of order , find the order of the matrix , if possible.

• A
• B
• C
• Dundefined
• E

Q24:

Suppose Which of the following products is defined?

• A
• B
• C
• D
• E

Q25:

Suppose is a matrix, is a matrix, and is a matrix. What are the sizes of the product matrices , and ?

• A
• B
• C
• D
• E