Answer to Question #117798 in Algebra for mike

Question

Number of pizzas sold

Size Mon. Tues. Wed. Thu. Fri. Sat. Sun.
Small 28 35 28 36 37 34 45
Mediu
m

36 37 47 32 36 38 47
Large 41 42 40 35 51 56 61
Family 30 33 39 33 78 76 83

a) Write the information in the table above as a 4 × 7 matrix, P.

b) Write down a suitable matrix, A, that when P is multiplied by A the resulting matrix will give the
number of each size of pizza sold in the week.

c) Calculate the matrix PA.


d) Write down a suitable matrix, B, that when B is multiplied by P the resulting matrix will give the
number of pizzas sold on each day of the week.

e) Calculate the matrix BP.

Answer 1

a) $P = \begin{pmatrix} 28 &35& 28 &36 &37 &34 &45 \\ 36 &37 &47 &32 &36& 38& 47\\ 41 &42 &40 &35 &51 &56 &61\\ 30 &33& 39 &33& 78& 76& 83 \end{pmatrix}$

b) $A=\begin{pmatrix} 1 \\ 1\\ 1\\ 1\\ 1\\ 1\\ 1 \end{pmatrix}$

c) $PA=\begin{pmatrix} PA_{11} \\ PA_{21}\\ PA_{31}\\ PA_{41} \end{pmatrix}$

PA₁₁ = P₁₁ * A₁₁ + P₁₂ * A₂₁ + P₁₃ * A₃₁ + P₁₄ * A₄₁ + P₁₅ * A₅₁ + P₁₆ * A₆₁ + P₁₇ * A₇₁ =

= 28 * 1 + 35 * 1 + 28 * 1 + 36 * 1 + 37 * 1 + 34 * 1 + 45 * 1 = 243

PA₂₁ = P₂₁ * A₁₁ + P₂₂ * A₂₁ + P₂₃ * A₃₁ + P₂₄ * A₄₁ + P₂₅ * A₅₁ + P₂₆ * A₆₁ + P₂₇ * A₇₁ =

= 36 * 1 + 37 * 1 + 47 * 1 + 32 * 1 + 36 * 1 + 38 * 1 + 47 * 1 = 273

PA₃₁ = P₃₁ * A₁₁ + P₃₂ * A₂₁ + P₃₃ * A₃₁ + P₃₄ * A₄₁ + P₃₅ * A₅₁ + P₃₆ * A₆₁ + P₃₇ * A₇₁ =

= 41 * 1 + 42 * 1 + 40 * 1 + 35 * 1 + 51 * 1 + 56 * 1 + 61 * 1 = 326

PA₄₁ = P₄₁ * A₁₁ + P₄₂ * A₂₁ + P₄₃ * A₃₁ + P₄₄ * A₄₁ + P₄₅ * A₅₁ + P₄₆ * A₆₁ + P₄₇ * A₇₁ =

= 30 * 1 + 33 * 1 + 39 * 1 + 33 * 1 + 78 * 1 + 76 * 1 + 83 * 1 = 372

$PA=\begin{pmatrix} 243 \\ 273\\ 326\\ 372 \end{pmatrix}$

d) $B=\begin{pmatrix} 1 & 1 & 1 &1 \end{pmatrix}$

e) BP = (BP₁₁ BP₁₂ BP₁₃ BP₁₄ BP₁₅ BP₁₆ BP₁₇)

BP₁₁ = B₁₁ * P₁₁ + B₁₂ * P₂₁ + B₁₃ * P₃₁ + B₁₄ * P₄₁ = 1 * 28 + 1 * 36 + 1 * 41 + 1 * 30 = 135

BP₁₂ = B₁₁ * P₁₂ + B₁₂ * P₂₂ + B₁₃ * P₃₂ + B₁₄ * P₄₂ = 1 * 35 + 1 * 37 + 1 * 42 + 1 * 33 = 147

BP₁₃ = B₁₁ * P₁₃ + B₁₂ * P₂₃ + B₁₃ * P₃₃ + B₁₄ * P₄₃ = 1 * 28 + 1 * 47 + 1 * 40 + 1 * 39 = 154

BP₁₄ = B₁₁ * P₁₄ + B₁₂ * P₂₄ + B₁₃ * P₃₄ + B₁₄ * P₄₄ = 1 * 36 + 1 * 32 + 1 * 35 + 1 * 33 = 136

BP₁₅ = B₁₁ * P₁₅ + B₁₂ * P₂₅ + B₁₃ * P₃₅ + B₁₄ * P₄₅ = 1 * 37 + 1 * 36 + 1 * 51 + 1 * 78 = 202

BP₁₆ = B₁₁ * P₁₆ + B₁₂ * P₂₆ + B₁₃ * P₃₆ + B₁₄ * P₄₆ = 1 * 34 + 1 * 38 + 1 * 56 + 1 * 76 = 204

BP₁₇ = B₁₁ * P₁₇ + B₁₂ * P₂₇ + B₁₃ * P₃₇ + B₁₄ * P₄₇ = 1 * 45 + 1 * 47 + 1 * 61 + 1 * 83 = 236

BP = (135 147 154 136 202 204 236)