# Worksheet: Properties of Matrix Multiplication

In this worksheet, we will practice identifying the properties of the multiplication of matrices and comparing them to the properties of multiplication of numbers.

**Q2: **

Matrices , and are square matrices. Use the associative law for three square matrices to determine which of the following proves that .

- A
- B
- C
- D

**Q3: **

Consider the matrices and . Is ?

- ANo
- BYes

**Q4: **

Given the matrices and , is ?

- Ayes
- Bno

**Q6: **

Given that matrices and , is ?

- Ayes
- Bno

**Q7: **

State whether the following statement is true or false: If and are both matrices, then is never the same as .

- Afalse
- Btrue

**Q8: **

Is there a matrix , other than the identity matrix , where for every matrix ?

- AYes
- BNo

**Q9: **

Given three matrices and , which of the following is equivalent to ?

- A
- B
- C
- D
- E

**Q10: **

State whether the following statement is true or false: If is a matrix and and are matrices, then .

- Atrue
- Bfalse

**Q11: **

Suppose that , and .

Find .

- A
- B
- C
- D
- E

Find .

- A
- B
- C
- D
- E

Find .

- A
- B
- C
- D
- E

Express in terms of and .

- A
- B
- C
- D
- E

**Q12: **

Given that is it true that ?

- Ano
- Byes

**Q14: **

Given that and is the identity matrix of the same order as , find and .

- A ,
- B ,
- C ,
- D ,

**Q17: **

Let , and

Find .

- A
- B
- C
- D
- E

Find .

- A
- B
- C
- D
- E

Find .

- A
- B
- C
- D
- E

Find .

- A
- B
- C
- D
- E

**Q18: **

What is the value of for any matrix ?

- A
- B
- C
- D

**Q19: **

Let be a matrix whose entries are all zero. If is any matrix and is any matrix, which of following is equivalent to ?

- A
- B
- C
- D
- E

**Q20: **

Given that is it true that ?

- Ano
- Byes

**Q21: **

Let and be the identity matrix. Find , , and their product , and then use this to express as a combination of and .

- A , , , .
- B , , , .
- C , , , .
- D , , , .
- E , , , .

**Q22: **

If and , is ?

- Ayes
- Bno

**Q23: **

Suppose that , , and .

Find .

- A
- B
- C
- D
- E

Find .

- A
- B
- C
- D
- E

Find .

- A
- B
- C
- D
- E

Express in terms of and .

- A
- B
- C
- D
- E

**Q24: **

and are two matrices with the property that for any matrix , and . Are and equal?

- ANo, they are different matrices of the same dimensions.
- BNo, they have different dimensions.
- CYes, they are both the identity matrix.

**Q25: **

and are two matrices with the property that for any matrix , and . Are and equal?

- AYes, they are both the identity matrix.
- BNo, they have different dimensions.
- CNo, they are different matrices of the same dimensions.