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Common Core: High School - Number and Quantity : Understanding the Multiplication Concept in Matrices as the Associative and Distributive Properties: CCSS.Math.Content.HSN-VM.C.9

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Example Questions

Example Question #1 : Understanding The Multiplication Concept In Matrices As The Associative And Distributive Properties: Ccss.Math.Content.Hsn Vm.C.9

Which of the following properties does not apply to matrices?

Possible Answers:

Distributive

Commutative

Associative

None of the answers

Correct answer:

Commutative

Explanation:

Commutative does not apply to matrices because if we have matrices , and . It is not necessarily true that $A\times B=B\times A$ , even though in some cases it's true.

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Example Question #2 : Understanding The Multiplication Concept In Matrices As The Associative And Distributive Properties: Ccss.Math.Content.Hsn Vm.C.9

Which is an example of two matrices satisfying the associative and distributive properties? Let a be a scalar, and A, B, and C be three unique matrices.

Possible Answers:

A+(B+C)=(A+B)+C

a(A+B)+C=aA+(aB+C)

$A\times B=B\times A$

a(A+B)=aA+aB

Correct answer:

a(A+B)+C=aA+(aB+C)

Explanation:

a(A+B)+C=aA+(aB+C) is the correct answer because it is the only answer that involves both the associative and distributive properties.

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Example Question #3 : Understanding The Multiplication Concept In Matrices As The Associative And Distributive Properties: Ccss.Math.Content.Hsn Vm.C.9

Which matrix when multiplied with

$\begin{bmatrix} 1&3 \\ 5& -4 \end{bmatrix}$

will yield the same result regardless of the order in which they're multiplied?

Possible Answers:

$\begin{bmatrix} 1&1 \\ 1& 1 \end{bmatrix}$

$\begin{bmatrix} 1&2 \\ 3& 4 \end{bmatrix}$

$\begin{bmatrix} 10&-10 \\ -2& 3 \end{bmatrix}$

$\begin{bmatrix} 1&4 \\ 4& 1 \end{bmatrix}$

$\begin{bmatrix} 0&0 \\ 0& 0 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 0&0 \\ 0& 0 \end{bmatrix}$

Explanation:

The only matrix that works is $\begin{bmatrix} 0&0 \\ 0& 0 \end{bmatrix}$ , because regardless of the order of matrix multiplication, the result will always be $\begin{bmatrix} 0&0 \\ 0& 0 \end{bmatrix}$ .

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Example Question #3 : Understanding The Multiplication Concept In Matrices As The Associative And Distributive Properties: Ccss.Math.Content.Hsn Vm.C.9

Why doesn't the commutative property hold for matrix multiplication?

Possible Answers:

Commutative property doesn't work for regular multiplication

Order of multiplication matters

Correct answer:

Order of multiplication matters

Explanation:

The reason that the commutative property doesn't apply to matrix multiplication is because order of multiplication matters. We multiply by the entry in the row of the first matrix by the entry in the column of the second matrix.

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Example Question #3 : Understanding The Multiplication Concept In Matrices As The Associative And Distributive Properties: Ccss.Math.Content.Hsn Vm.C.9

Which is an example of two matrices satisfying the distributive properties? Let be a scalar, and ,, and be three unique matrices.

Possible Answers:

a(A+B)=aA+aB

(A+B)+C=A+(B+C)

$A\times B=B\times A$

Correct answer:

a(A+B)=aA+aB

Explanation:

The only answer that satisfies the distributive property is a(A+B)=aA+aB .

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Example Question #4 : Understanding The Multiplication Concept In Matrices As The Associative And Distributive Properties: Ccss.Math.Content.Hsn Vm.C.9

True or False: If A, B, and are square matrices, is (AB)C=A(BC) ?

Possible Answers:

True

False

Correct answer:

True

Explanation:

Let

$A=\begin{bmatrix} a_1& a_2\\ a_3&a_4 \end{bmatrix}$ , $B=\begin{bmatrix} b_1& b_2\\ b_3&b_4 \end{bmatrix}$ , $C=\begin{bmatrix} c_1& c_2\\ c_3&c_4 \end{bmatrix}$ .

$(AB)C=\left(\begin{bmatrix} a_1& a_2\\ a_3&a_4 \end{bmatrix}\begin{bmatrix} b_1& b_2\\ b_3&b_4 \end{bmatrix} \right )\begin{bmatrix} c_1&c_2 \\ c_3& c_4 \end{bmatrix}$

Now do matrix multiplication inside the parenthesis.

$\left(\begin{bmatrix} a_1& a_2\\ a_3&a_4 \end{bmatrix}\begin{bmatrix} b_1& b_2\\ b_3&b_4 \end{bmatrix} \right )\begin{bmatrix} c_1&c_2 \\ c_3& c_4 \end{bmatrix} =\begin{bmatrix} a_1b_1+a_2b_3 &a_1b_2+a_2b_4 \\ a_3b_1+a_4b_3&a_3b_2+a_4b_4 \end{bmatrix} \begin{bmatrix} c_1&c_2 \\ c_3&c_4 \end{bmatrix}$

Now multiply the result by the other matrix to get

$=\begin{bmatrix} c_1(a_1b_1+a_2b_3)+c_3(a_1b_2+a_2b_4)& c_2(a_1b_1+a_2b_3)+c_4(a_1b_2+a_2b_4) \\ c_1(a_3b_1+a_4b_3)+c_3(a_3b_2+a_4b_4)& c_2(a_3b_1+a_4b_3)+c_4(a_3b_2+a_4b_4) \end{bmatrix}$

Now lets do it from the other side

$A(BC)=\begin{bmatrix} a_1& a_2\\ a_3&a_4 \end{bmatrix}\left(\begin{bmatrix} b_1& b_2\\ b_3&b_4 \end{bmatrix} \begin{bmatrix} c_1&c_2 \\ c_3& c_4 \end{bmatrix}\right )$

Do the matrix multiplication inside the parenthesis first

$\begin{bmatrix} a_1& a_2\\ a_3&a_4 \end{bmatrix}\left(\begin{bmatrix} b_1& b_2\\ b_3&b_4 \end{bmatrix} \begin{bmatrix} c_1&c_2 \\ c_3& c_4 \end{bmatrix}\right )=\begin{bmatrix} a_1& a_2\\ a_3&a_4 \end{bmatrix}\begin{bmatrix} b_1c_1+b_2c_3&b_1c_2+b_2c_4 \\ b_3c_1+b_4c_3&b_3c_2+b_4c_4 \end{bmatrix}$

Now multiply the result by the other matrix to get

$\begin{bmatrix} a_1& a_2\\ a_3&a_4 \end{bmatrix}\begin{bmatrix} b_1c_1+b_2c_3&b_1c_2+b_2c_4 \\ b_3c_1+b_4c_3&b_3c_2+b_4c_4 \end{bmatrix}=\begin{bmatrix} a_1(b_1c_1+b_2c_3)+a_2(b_3c_1+b_4c_3)&a_1(b_1c_2+b_2c_4)+a_2(b_3c_2+b_4c_4) \\ a_3(b_1c_1+b_2c_3)+a_4(b_3c_1+b_4c_3)& a_3(b_1c_2+b_2c_4)+a_4(b_3c_2+b_4c_4) \end{bmatrix}$

If we rearrange the terms in this matrix we get

$=\begin{bmatrix} c_1(a_1b_1+a_2b_3)+c_3(a_1b_2+a_2b_4)& c_2(a_1b_1+a_2b_3)+c_4(a_1b_2+a_2b_4) \\ c_1(a_3b_1+a_4b_3)+c_3(a_3b_2+a_4b_4)& c_2(a_3b_1+a_4b_3)+c_4(a_3b_2+a_4b_4) \end{bmatrix}$

Since these are the same matrix, we have evidence that the statement is true, (AB)C=A(BC) .

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Example Question #7 : Understanding The Multiplication Concept In Matrices As The Associative And Distributive Properties: Ccss.Math.Content.Hsn Vm.C.9

True or False: The following matrix product is possible.

$\begin{bmatrix} 3& 2 \end{bmatrix}\begin{bmatrix} 3& 4 \end{bmatrix}$

Possible Answers:

True

False

Correct answer:

False

Explanation:

The answer is false because the dimensions are $1\times 2$ for each matrix.

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Example Question #4 : Understanding The Multiplication Concept In Matrices As The Associative And Distributive Properties: Ccss.Math.Content.Hsn Vm.C.9

True or False:

The following matrix multiplication is possible.

$\begin{bmatrix} 4&3 &2 \end{bmatrix} \begin{bmatrix} 6\\ 7\\ 8 \end{bmatrix}$

Possible Answers:

False

True

Correct answer:

True

Explanation:

The matrix multiplication is possible since the dimensions will work out. The result will be a $3\times 3$ since the dimensions are $3\times1$ , and $1\times 3$ .

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Example Question #5 : Understanding The Multiplication Concept In Matrices As The Associative And Distributive Properties: Ccss.Math.Content.Hsn Vm.C.9

True or False:

The following matrix multiplication is possible

$\begin{bmatrix} 3 \end{bmatrix} \begin{bmatrix} 4&5 &6 &7 &8 \end{bmatrix}$

Possible Answers:

False

True

Correct answer:

True

Explanation:

The matrix multiplication is possible because the dimensions work out. The resulting matrix will be $1\times 5$ , because the matrices are $1\times1$ , and $1\times 5$ .

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Example Question #6 : Understanding The Multiplication Concept In Matrices As The Associative And Distributive Properties: Ccss.Math.Content.Hsn Vm.C.9

True or False:

The following matrix multiplication is possible.

$\begin{bmatrix} 4& 5& 6& 7\\ 9& 0& 5&4 \end{bmatrix} \begin{bmatrix} 3& 4& 6& 7\\ 34& 76& 9& 90\\ 1& 3& 4& 67 \end{bmatrix}$

Possible Answers:

True

False

Correct answer:

False

Explanation:

The matrix multiplication is not possible because the dimensions do not work out. You can't multiply a $2\times 4$ and a $3\times4$ matrix together.

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Common Core: High School - Number and Quantity : Understanding the Multiplication Concept in Matrices as the Associative and Distributive Properties: CCSS.Math.Content.HSN-VM.C.9

Study concepts, example questions & explanations for Common Core: High School - Number and Quantity

All Common Core: High School - Number and Quantity Resources

Example Questions

Example Question #1 : Understanding The Multiplication Concept In Matrices As The Associative And Distributive Properties: Ccss.Math.Content.Hsn Vm.C.9

Example Question #2 : Understanding The Multiplication Concept In Matrices As The Associative And Distributive Properties: Ccss.Math.Content.Hsn Vm.C.9

Example Question #3 : Understanding The Multiplication Concept In Matrices As The Associative And Distributive Properties: Ccss.Math.Content.Hsn Vm.C.9

Example Question #3 : Understanding The Multiplication Concept In Matrices As The Associative And Distributive Properties: Ccss.Math.Content.Hsn Vm.C.9

Example Question #3 : Understanding The Multiplication Concept In Matrices As The Associative And Distributive Properties: Ccss.Math.Content.Hsn Vm.C.9

Example Question #4 : Understanding The Multiplication Concept In Matrices As The Associative And Distributive Properties: Ccss.Math.Content.Hsn Vm.C.9

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Example Question #4 : Understanding The Multiplication Concept In Matrices As The Associative And Distributive Properties: Ccss.Math.Content.Hsn Vm.C.9

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All Common Core: High School - Number and Quantity Resources

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