Chapter 11 Exercises: Matrix Operations

Consider the following matrices:

\[ \mathbf{X} = \begin{bmatrix}2 & 3 \\1 & 2 \end{bmatrix} \qquad \mathbf{Y} = \begin{bmatrix}3 & 4 \\2 & 1 \end{bmatrix} \qquad \mathbf{Z} = \begin{bmatrix}2 & 3 & 1 \\5 & 6 & 8\\9 & 4 & 7 \end{bmatrix} \]

1. Find \(\mathbf{X} + \mathbf{Y}\).

\[ \begin{split} \mathbf{X}+\mathbf{Y} &= \begin{bmatrix}2+3 & 3+4 \\1+2 & 2+1 \end{bmatrix} \\[1em] &= \begin{bmatrix}5 & 7 \\3 & 3 \end{bmatrix} \end{split} \]

2. Find \(\mathbf{X} - \mathbf{Y}\).

\[ \begin{split} \mathbf{X}-\mathbf{Y} &= \begin{bmatrix}2-3 & 3-4 \\1-2 & 2-1 \end{bmatrix} \\[1em] &= \begin{bmatrix}-1 & -1 \\-1 & 1 \end{bmatrix} \end{split} \]

3. Find \(3 \mathbf{Z}\).

\[ \begin{split} 3\mathbf{Z} &= \begin{bmatrix}3(2) & 3(3) & 3(1) \\3(5) & 3(6) & 3(8)\\3(9) & 3(4) & 3(7) \end{bmatrix} \\[1em] &= \begin{bmatrix}6 & 9 & 3 \\15 & 18 & 24\\27 & 12 & 21 \end{bmatrix} \end{split} \]

4. Find \(-2 \mathbf{X} + 4\mathbf{Y}\).

\[ \begin{split} -2\mathbf{X}+4\mathbf{Y} &= \begin{bmatrix}-2(2)+4(3) & -2(3)+4(4) \\-2(1)+4(2) & -2(2)+4(1) \end{bmatrix} \\[1em] &= \begin{bmatrix}8 & 10 \\6 & 0 \end{bmatrix} \end{split} \]

Consider the following matrices:

\[ \mathbf{A} = \begin{bmatrix}0 & 6 \\5 & 1 \end{bmatrix} \qquad \mathbf{B} = \begin{bmatrix}0 & 5 \\2 & \frac{1}{2} \end{bmatrix} \qquad \mathbf{C} = \begin{bmatrix}6 & 2 & 1 \\5 & 3 & 1\\8 & 4 & 1 \end{bmatrix} \qquad \mathbf{D} = \begin{bmatrix}0& 1 & 4 & 6 \\1 & 2 & 5 & -2\\1 & 3 & 2 & 8 \end{bmatrix} \]

5. Is AB conformable?

Yes. Since A has two columns and B has the same number of rows, AB is conformable.

\[ \underset{2\times\color{red}{2}}{\mathbf{A}}~\underset{{\color{red}{2}} \times2}{\mathbf{B}} \]

6. Is BC conformable?

No. Since B has two columns and C has three rows, BC is not conformable.

\[ \underset{2\times\color{red}{2}}{\mathbf{B}}~\underset{{\color{red}{3}} \times3}{\mathbf{C}} \]

7. Is the product where we premultiply D by C conformable?

Yes. Since C has three columns and D has the same number of rows, CD is conformable.

\[ \underset{3\times\color{red}{3}}{\mathbf{C}}~\underset{{\color{red}{3}} \times4}{\mathbf{D}} \]

Statistics Example: Weights

8. Consider the scores for five students on four course exams (each out of 100 points) shown in matrix X. The final percentage in the course is based on the following weighting: the first and second exams are worth 10% of the course, the third exam is worth 30% of the course, and the fourth exam is worth 50% of the course. These weights are presented in the column vector w. Use R to find the final percentage for each student by postmultiplying the score matrix by the weight vector.

\[ \mathbf{X} = \begin{bmatrix}32 & 54 & 56 & 21 \\42 & 23 & 52 & 35 \\ 16 & 41 & 54 & 56 \\ 58 & 52 & 31 & 24 \\ 41 & 50 & 42 & 40 \end{bmatrix} \qquad \mathbf{w} = \begin{bmatrix} 0.10 \\ 0.10 \\ 0.30 \\ 0.50 \end{bmatrix} \]

# Create X
X = matrix(
  data = c(32, 54, 56, 21,
           42, 23, 52, 35,
           16, 41, 54, 56,
           58, 52, 31, 24,
           41, 50, 42, 40),
  byrow = TRUE, 
  ncol = 4
)

# Create w
w = matrix(data = c(0.10, 0.10, 0.30, 0.50), ncol = 1)

# Compute Xw
X %*% w

     [,1]
[1,] 35.9
[2,] 39.6
[3,] 49.9
[4,] 32.3
[5,] 41.7