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In this lesson, we will learn how to use the properties of matrix multiplication.

Q1:

Given that find π΄ π΅ and π΅ π΄ .

Q2:

Matrices π΄ , π΅ , πΆ , and π· are square matrices. Use the associative law for three square matrices to determine which of the following proves that π΄ ( π΅ ( πΆ π· ) ) = ( ( π΄ π΅ ) πΆ ) π· .

Q3:

Consider the matrices Is ?

Q4:

Are the following matrices multiplicative inverses of each other?

Q5:

Given the 1 Γ 1 matrices π΄ = [ 3 ] and π΅ = [ 4 ] , is π΄ π΅ = π΅ π΄ ?

Q6:

Given the matrices and , is ?

Q7:

Given that matrices and , is ?

Q8:

State whether the following statement is true or false: If π΄ and π΅ are both 2 Γ 2 matrices, then π΄ π΅ is never the same as π΅ π΄ .

Q9:

Is there a 2 Γ 2 matrix π΄ , other than the indentity matrix πΌ , where π΄ π = π π΄ for every 2 Γ 2 matrix π ?

Q10:

Given three matrices π΄ , π΅ , and πΆ , which of the following is equivalent to π΄ ( π΅ + πΆ ) ?

Q11:

State whether the following statement is true or false: If π΄ is a 2 Γ 3 matrix and π΅ and πΆ are 3 Γ 2 matrices, then π΄ ( π΅ + πΆ ) = π΄ πΆ + π΄ π΅ .

Q12:

Suppose that , and .

Find .

Express in terms of and .

Q13:

Given that is it true that ( π΄ π΅ ) πΆ = π΄ ( π΅ πΆ ) ?

Q14:

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

Q15:

Consider the matrices By setting equal to the zero matrix, find matrices and such that if , then for some numbers , and .

Q16:

Suppose π΄ π΅ = π΄ πΆ and π΄ is an invertible π Γ π matrix. Does it follow that π΅ = πΆ ?

Q17:

Given that π΄ = οΌ β 1 4 β 1 1 1 ο and πΌ is the identity matrix of the same order as π΄ , find π΄ Γ πΌ and πΌ 2 .

Q18:

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

Q19:

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

Q20:

Let , and

Q21:

What is the value of π΄ + ( β π΄ ) for any matrix π΄ ?

Q22:

If the matrices π΄ and π΅ both have order π Γ π , then what is the order of the matrix π΄ β 2 π΅ ?

Q23:

Find a matrix such that for all matrices .

Q24:

Let π be a 2 Γ 3 matrix whose entries are all zero. If π΄ is any 2 Γ 3 matrix and π΅ is any 2 Γ 2 matrix, which of following is equivalent to π΄ + π΅ π ?

Q25:

Given that is it true that π΄ ( π΅ + πΆ ) = π΄ π΅ + π΄ πΆ ?

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