# Exploring Matrices Exercises

Try you hand at a few exercises involving Exploring Matrices.

### Contents

#### Qube Simplified

I have simplified the `Qube` app by removing these four buttons.

`solve`. The <== key now controls the unscrambling operation.

`scramble`. The ==> key now does six random rotations.

`order`. I never found a satisfactory reference for the group theory of Rubik's cube.

`score`. I never found a use for the nuclear norm.

Code for `Qube` dated 9/24/2023 is included in the Apps mzip archive.

#### Exploring Matrices Exercises

Here are a few exercises for Exploring Matrices. The answers are available at ExMatAnswers.

#### Matrix Multiplication

**1.** Compute by rows, and by columns.

$$ \left( \begin{array}{rrr} 8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2 \end{array} \right) \left( \begin{array}{r} 1 \\ 1 \\ 1 \end{array} \right) $$

**2.** Solve for $z$ using inner products of rows, and using linear combinations of columns.

$$ \left( \begin{array}{rrr} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} \right) \left( \begin{array}{r} 1 \\ z \\ 1 \end{array} \right) \ = \ \left( \begin{array}{r} 0 \\ 0 \\ 0 \end{array} \right) $$

**3.** What do the `m`, `n` and `p` buttons on the `Multiply` app do? What are the other buttons and what do they do?

**4.** If *A* is n-by-n and *x* is n-by-1, how many multiplications are required to compute *A x* ?

**5.** If *A* is m-by-n and *B* is n-by-p, how many multiplications are required to compute *A B* ?

#### Rotations and Scaling

**1.** What is *R*(30º)?

$$ R(\theta) \ = \ \left( \begin{array}{rr} \cos{\theta} & \sin{\theta} \\ -\sin{\theta} & \cos{\theta} \end{array} \right) $$

**2.** Explain https://xkcd.com/184.

**3.** What is the value of $\theta$ ?

$$ R(\theta) \ = \ \left( \begin{array}{rr} 0.8 & 0.6 \\ -0.6 & 0.8 \end{array} \right) $$

**4.** What is the value of $\theta$ ?

**5.** Edit a copy of `Rotate.m` and replace the house with a hand. You can use my hand or your own hand; see exercise 3.4 in Numerical Computing with MATLAB .

#### Computer Graphics

**1.** Show how homogeneous coordinates and matrix-vector multiplication by `Tx`, `Ty` or `Tz` produce translation.

**2.** What is the range of the rotations used by the pitch, roll, and yaw buttons on the `Grafix` app?

**3.** What color in the beacon on top of the plane? How would you change the beacon's color?

**4,** What is the function of the resolution and offset sliders for the teapot?

**5,** How many times does the bucky ball bounce off the sides of the plot window?

#### Matrices and Cubes

**1.** What is the color of central cubelet in the Color Cube?

**2.** What do the "<=" , "<==" , "=>" and "==>" buttons on `Qube` do?

**3.** What is "God's Number" for a 3-by-3-by-3 Rubik's Cube? What are `Q20` and `Q26`? See Cleve's Corner 2022/09/05.

**4.** Can you restore the following scrambled cubes with fewer moves than `<==`, the unscramble key? Use the quarter-turn metric and reset the cube with `start` or by clicking on `stack` and `Q0`. You might also want to set `speed` to 30 or 45,

`LRUDFB`

`LRL'R'`

`FLLLRB`

`Q26`

- Reset the random number generator by entering
`rng(r)`for some small integer`r`in the command window and then generate six random rotations with the`==>`key.

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