Lesson test: Matrices and their product

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Matrices and their product

1.[2p]
In the product $A B$ , which transformation acts on a vector first?

$B$

Correct

$A$

The answer is: $B$

Neither, since the order does not matter

The answer is: $B$

Whichever has more columns

The answer is: $B$

2.[2p]
Column $j$ of $A B$ is

$A$ applied to column $j$ of $B$

Correct

$B$ applied to column $j$ of $A$

The answer is: $A$ applied to column $j$ of $B$

the dot product of column $j$ of $A$ with column $j$ of $B$

The answer is: $A$ applied to column $j$ of $B$

the sum of column $j$ of $A$ and column $j$ of $B$

The answer is: $A$ applied to column $j$ of $B$

3.[3p]
With $A = [(2, 1), (- 1, 3)]$ and $B = [(1, 4), (2, 0)]$ , what is the entry in row 2, column 1 of $A B$ ?

CorrectNot quite: 13

4.[1p]
For all square matrices of the same size, $A B = B A$ .

True

The answer is: False

False

Correct

5.[3p]
Which of these hold for all matrices of compatible shapes?

Select all that apply

$(A B) C = A (B C)$

Correct

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

Correct

$(A B)^{T} = B^{T} A^{T}$

Correct

$A B = 0$ implies $A = 0$ or $B = 0$

The answer is: $(AB)C = A(BC)$, $A(B + C) = AB + AC$, $(AB)^T = B^TA^T$

6.[2p]
Squaring the shear $[(1, 0), (1, 1)]$ gives $[(1, 0), (k, 1)]$ . What is $k$ ?

CorrectNot quite: 2

7.[2p]
Why is $(A B) C = A (B C)$ true without any calculation?

Both sides describe doing $C$ , then $B$ , then $A$ , and composing functions is associative

Correct

Because matrix entries are real numbers and multiplication of reals is associative

The answer is: Both sides describe doing $C$, then $B$, then $A$, and composing functions is associative

Because the transpose reverses products

The answer is: Both sides describe doing $C$, then $B$, then $A$, and composing functions is associative

It is true only for square matrices

The answer is: Both sides describe doing $C$, then $B$, then $A$, and composing functions is associative

8.[2p]
Multiplying two $n$ by $n$ matrices by the definition costs about $n^{3}$ multiplications. For $n = 1000$ , how many is that, as a power of ten?

CorrectNot quite: 9

9.[3p]
Match each product to what it equals.

A quarter turn composed with itself
Projection onto the horizontal axis, squared
Projection onto the horizontal axis times projection onto the vertical
Any matrix times the identity

that matrix
a half turn, which is $- I$
the same projection
the zero matrix

Show the answer

A quarter turn composed with itself: a half turn, which is $- I$ Projection onto the horizontal axis, squared: the same projection Projection onto the horizontal axis times projection onto the vertical: the zero matrix Any matrix times the identity: that matrix

1.[2p] In the product AB, which transformation acts on a vector first?

2.[2p] Column j of AB is

3.[3p] With A=[(2,1),(-1,3)] and B=[(1,4),(2,0)], what is the entry in row 2, column 1 of AB?

4.[1p] For all square matrices of the same size, AB=BA.

5.[3p] Which of these hold for all matrices of compatible shapes?

6.[2p] Squaring the shear [(1,0),(1,1)] gives [(1,0),(k,1)]. What is k?

7.[2p] Why is (AB)C=A(BC) true without any calculation?

8.[2p] Multiplying two n by n matrices by the definition costs about n3 multiplications. For n=1000, how many is that, as a power of ten?

9.[3p] Match each product to what it equals.

1.[2p]
In the product $A B$ , which transformation acts on a vector first?

2.[2p]
Column $j$ of $A B$ is

3.[3p]
With $A = [(2, 1), (- 1, 3)]$ and $B = [(1, 4), (2, 0)]$ , what is the entry in row 2, column 1 of $A B$ ?

4.[1p]
For all square matrices of the same size, $A B = B A$ .

5.[3p]
Which of these hold for all matrices of compatible shapes?

6.[2p]
Squaring the shear $[(1, 0), (1, 1)]$ gives $[(1, 0), (k, 1)]$ . What is $k$ ?

7.[2p]
Why is $(A B) C = A (B C)$ true without any calculation?

8.[2p]
Multiplying two $n$ by $n$ matrices by the definition costs about $n^{3}$ multiplications. For $n = 1000$ , how many is that, as a power of ten?

9.[3p]
Match each product to what it equals.