# 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.

Q1:

Given that find and .

• A,
• B,
• C,
• D,

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

Q5:

Given the matrices and , is ?

• Ano
• Byes

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

Q13:

From the following, choose two matrices, and , such that , , and .

• A,
• B,
• C,
• D,
• E,

Q14:

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

• A,
• B,
• C,
• D,

Q15:

From the following, choose two matrices, and , such that , with .

• A,
• B,
• C,
• D,
• E,

Q16:

If and are symmetric matrices, then the product is also symmetric only when and are .

• Amatrices that commute
• BHermitian matrices
• Csquare matrices
• Dinvertible matrices

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.