## Engage NY Eureka Math Precalculus Module 2 Lesson 11 Answer Key

### Eureka Math Precalculus Module 2 Lesson 11 Exercise Answer Key

Opening Exercise

Kiamba thinks A+B=B+A for all 2×2 matrices. Rachel thinks it is not always true. Who is correct? Explain.

Answer:

Let A=\(\left(\begin{array}{ll}

a_{11} & a_{12} \\

a_{21} & a_{22}

\end{array}\right)\) and B=\(\left(\begin{array}{ll}

\boldsymbol{b}_{11} & \boldsymbol{b}_{12} \\

\boldsymbol{b}_{21} & \boldsymbol{b}_{22}

\end{array}\right)\)

What is the sum of A+B?

The two matrices must be equal because each of the sums must be equal according to the commutative property of addition of real numbers. Kiamba is correct.

Exercises

Exercise 1.

In two-dimensional space, let A be the matrix representing a rotation about the origin through an angle of 45°, and let B be the matrix representing a reflection about the x-axis. Let x be the point \(\left(\begin{array}{l}

1 \\

1

\end{array}\right)\).

a. Write down the matrices A, B, and A+B.

Answer:

b. Write down the image points of Ax, Bx, and (A+B)x, and plot them on graph paper.

Answer:

Ax=\(\left(\begin{array}{c}

\mathbf{0} \\

\sqrt{2}

\end{array}\right)\) Bx=\(\left(\begin{array}{c}

1 \\

-1

\end{array}\right)\) (A+B)x=\(\left(\begin{array}{c}

1 \\

\sqrt{2}-1

\end{array}\right)\)

c. What do you notice about (A+B)x compared to Ax and Bx?

Answer:

The point (A+B)x is equal to the sum of the points Ax and Bx by the distributive property.

Exercise 2.

For three matrices of equal size, A, B, and C, does it follow that A+(B+C)=(A+B)+C?

a. Determine if the statement is true geometrically. Let A be the matrix representing a reflection across the y-axis. Let B be the matrix representing a counterclockwise rotation of 30°. Let C be the matrix representing a reflection about the x-axis. Let x be the point \(\left(\begin{array}{l}

1 \\

1

\end{array}\right)\).

Answer:

From the graph, we see that Ax+(Bx+Cx)=(Ax+Bx)+Cx.

b. Confirm your results algebraically.

Answer:

c. What do your results say about matrix addition?

Answer:

Matrix addition is associative.

Exercise 3.

If x=\(\left(\begin{array}{l}

\boldsymbol{x} \\

\boldsymbol{y} \\

\boldsymbol{Z}

\end{array}\right)\), what are the coordinates of a point y with the property x+y if the origin O=\(\left(\begin{array}{l}

0 \\

0 \\

0

\end{array}\right) ?\)

Answer:

y=\(\left(\begin{array}{c}

-x \\

-y \\

-z

\end{array}\right)\)

Exercise 4.

Suppose A=\(\left(\begin{array}{ccc}

11 & -5 & 2 \\

-34 & 6 & 19 \\

8 & -542 & 0

\end{array}\right)\) and matrix B has the property that Ax+Bx is the origin. What is the matrix B?

Answer:

B=\(\left(\begin{array}{ccc}

-11 & 5 & -2 \\

34 & -6 & -19 \\

-8 & 542 & 0

\end{array}\right)\)

Exercise 5.

For three matrices of equal size, A, B, and C, where A represents a reflection across the line y=x, B represents a counterclockwise rotation of 45°, C represents a reflection across the y-axis, and x=\(\left(\begin{array}{l}

1 \\

2

\end{array}\right)\):

a. Show that matrix addition is commutative: Ax+Bx=Bx+Ax.

Answer:

b. Show that matrix addition is associative: Ax+(Bx+Cx)=(Ax+Bx)+Cx.

Answer:

Exercise 6.

Let A, B, C, and D be matrices of the same dimensions. Use the commutative property of addition of two matrices to prove A+B+C=C+B+A.

Answer:

If we treat A+B as one matrix, then:

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

=C+(B+A)

=C+B+A

### Eureka Math Precalculus Module 2 Lesson 11 Problem Set Answer Key

Question 1.

Let A be a matrix transformation representing a rotation of 45° about the origin and B be a reflection across the y-axis. Let x=(3,4).

a. Represent A and B as matrices, and find A+B.

Answer:

b. Represent Ax and Bx as matrices, and find (A+B)x.

Answer:

c. Graph your answer to part (b).

Answer:

See graph in part (d).

d. Draw the parallelogram containing Ax, Bx, and (A+B)x.

Answer:

Question 2.

Let A be a matrix transformation representing a rotation of 300° about the origin and B be a reflection across the

x-axis. Let x=(2,-5).

a. Represent A and B as matrices, and find A+B.

Answer:

b. Represent Ax and Bx as matrices, and find (A+B)x.

Answer:

c. Graph your answer to part (b).

Answer:

See graph in part (d).

d. Draw the parallelogram containing Ax, Bx, and (A+B)x.

Answer:

Question 3.

Let A, B, C, and D be matrices of the same dimensions.

a. Use the associative property of addition for three matrices to prove (A+B)+(C+D)=A+(B+C)+D.

Answer:

If we treat C+D as one matrix, then:

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

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

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

=A+(B+C)+D

b. Make an argument for the associative and commutative properties of addition of matrices to be true for finitely many matrices being added.

Answer:

For finitely many matrices, we can always use the formula that has been proven already and break the rest of the problem down into that many pieces, just like we did in part (a). Since this is true for any finite number, we could also use the formula that has been proven for one less than whatever number we are trying to prove the property true for.

Question 4.

Let A be a m×n matrix with element in the i^{th} row, j^{th} column a_{ij}, and B be a m×n matrix with element in the i^{th} row, j^{th} column b_{ij}. Present an argument that A+B=B+A.

Answer:

The entry in the i^{th} row, j^{th} column of A+B will be a_{ij}+b_{ij}, which is equal to b_{ij}+a_{ij}, which is the entry in the i^{th} row, j^{th} column of B+A. Since this is true for any element of A+B, we have that A+B=B+A for any two matrices of equal dimensions.

Question 5.

For integers x, y, define x⊕y=x⋅y+1; read “x plus y” where x⋅y is defined normally.

a. Is this form of addition commutative? Explain why or why not.

Answer:

Yes x⊕y=xy+1=yx+1=y⊕x

b. Is this form of addition associative? Explain why or why not.

Answer:

No

x⊕(y⊕z)=x⊕(yz+1)=x(yz+1)+1=xyz+x+1

(x⊕y)⊕z=(xy+1)⊕z=(xy+1)z+1=xyz+z+1

Question 6.

For integers x, y, define x⊕y=x.

a. Is this form of addition commutative? Explain why or why not.

Answer:

No x⊕y=x, and y⊕x=y

b. Is this form of addition associative? Explain why or why not.

Answer:

Yes x⊕(y⊕z)=x⊕y=x, and (x⊕y)⊕z=x⊕z=x

### Eureka Math Precalculus Module 2 Lesson 11 Exit Ticket Answer Key

Question 1.

Let x=\(\left(\begin{array}{l}

3 \\

-1

\end{array}\right)\), A=\(\left(\begin{array}{ll}

2 & 2 \\

2 & 0

\end{array}\right)\), and B=\(\left(\begin{array}{cc}

-1 & 0 \\

0 & -1

\end{array}\right)\).

a. Find and plot the points Ax, Bx, and (A+B)x on the axes below.

Answer:

b. Show algebraically that matrix addition is commutative: Ax+Bx=Bx+Ax.

Answer:

Ax=\(\left(\begin{array}{l}

4 \\

6

\end{array}\right)\), Bx=\(\left(\begin{array}{l}

-3 \\

1

\end{array}\right)\), Ax+Bx=\(\left(\begin{array}{l}

1 \\

7

\end{array}\right)\), and Bx+Ax=\(\left(\begin{array}{l}

1 \\

7

\end{array}\right)\) Ax+Bx=Bx+Ax

Question 2.

Let A=\(\left(\begin{array}{ll}

a & b \\

c & d

\end{array}\right)\) and B=\(\left(\begin{array}{ll}

x & y \\

z & w

\end{array}\right)\). Prove A+B=B+A.

Answer: