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We are given a matrix $C$ representing the sales of small, medium, and large coffees for one week at Faith's coffee shop. We are asked to: a) State the order of matrix $C$. b) Interpret the meaning of the element $C_{11}$. c) Multiply the matrix $[1\ 1\ 1]$ by matrix $C$ and explain the resulting information. d) Given matrix $D$ which represents the sales for the second week, calculate $C+D$.

AlgebraMatricesMatrix OperationsMatrix AdditionMatrix MultiplicationData Analysis
2025/4/27

1. Problem Description

We are given a matrix CC representing the sales of small, medium, and large coffees for one week at Faith's coffee shop. We are asked to:
a) State the order of matrix CC.
b) Interpret the meaning of the element C11C_{11}.
c) Multiply the matrix [1 1 1][1\ 1\ 1] by matrix CC and explain the resulting information.
d) Given matrix DD which represents the sales for the second week, calculate C+DC+D.

2. Solution Steps

a) The matrix CC has 3 rows (corresponding to Small, Medium, and Large sizes) and 7 columns (corresponding to Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, and Sunday). Therefore, the order of matrix CC is 3×73 \times 7.
b) The element C11C_{11} refers to the element in the first row and first column of matrix CC. This represents the number of small coffees sold on Monday. From the matrix CC, C11=24C_{11} = 24. So, C11C_{11} represents the number of small coffees sold on Monday.
c) We need to multiply the matrix [1 1 1][1\ 1\ 1] by matrix CC. This means we sum the entries in each column of CC.
[1 1 1]×C=[1 1 1]×[241817262125323223233128344015111412192631][1\ 1\ 1] \times C = [1\ 1\ 1] \times \begin{bmatrix} 24 & 18 & 17 & 26 & 21 & 25 & 32 \\ 32 & 23 & 23 & 31 & 28 & 34 & 40 \\ 15 & 11 & 14 & 12 & 19 & 26 & 31 \end{bmatrix}
The resulting matrix will be a 1×71 \times 7 matrix:
[24+32+15 18+23+11 17+23+14 26+31+12 21+28+19 25+34+26 32+40+31]=[71 52 54 69 68 85 103][24+32+15\ 18+23+11\ 17+23+14\ 26+31+12\ 21+28+19\ 25+34+26\ 32+40+31] = [71\ 52\ 54\ 69\ 68\ 85\ 103]
[71 52 54 69 68 85 103][71\ 52\ 54\ 69\ 68\ 85\ 103] provides the total number of coffees sold each day of the first week (Monday to Sunday).
d) We need to add matrix CC and matrix DD. Matrix DD is:
D=[222219222523342925253225343712121416202732]D = \begin{bmatrix} 22 & 22 & 19 & 22 & 25 & 23 & 34 \\ 29 & 25 & 25 & 32 & 25 & 34 & 37 \\ 12 & 12 & 14 & 16 & 20 & 27 & 32 \end{bmatrix}
C+D=[241817262125323223233128344015111412192631]+[222219222523342925253225343712121416202732]=[464036484648666148486353687727232828395363]C + D = \begin{bmatrix} 24 & 18 & 17 & 26 & 21 & 25 & 32 \\ 32 & 23 & 23 & 31 & 28 & 34 & 40 \\ 15 & 11 & 14 & 12 & 19 & 26 & 31 \end{bmatrix} + \begin{bmatrix} 22 & 22 & 19 & 22 & 25 & 23 & 34 \\ 29 & 25 & 25 & 32 & 25 & 34 & 37 \\ 12 & 12 & 14 & 16 & 20 & 27 & 32 \end{bmatrix} = \begin{bmatrix} 46 & 40 & 36 & 48 & 46 & 48 & 66 \\ 61 & 48 & 48 & 63 & 53 & 68 & 77 \\ 27 & 23 & 28 & 28 & 39 & 53 & 63 \end{bmatrix}

3. Final Answer

a) The order of matrix CC is 3×73 \times 7.
b) C11C_{11} represents the number of small coffees sold on Monday.
c) [1 1 1]×C=[71 52 54 69 68 85 103][1\ 1\ 1] \times C = [71\ 52\ 54\ 69\ 68\ 85\ 103]. This matrix provides the total number of coffees sold each day of the first week (Monday to Sunday).
d) C+D=[464036484648666148486353687727232828395363]C + D = \begin{bmatrix} 46 & 40 & 36 & 48 & 46 & 48 & 66 \\ 61 & 48 & 48 & 63 & 53 & 68 & 77 \\ 27 & 23 & 28 & 28 & 39 & 53 & 63 \end{bmatrix}

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