KJSEA 2025 Maths Q27 — Matrices: Forming and Adding Matrices

KJSEA 2025 Grade 9 Algebra

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The Question

“Two families, A and B, bought oranges, mangoes, and bananas over two weeks. Part (a): form matrices to represent this information. Part (b): find the total of each fruit bought by each family across the two weeks.”

1

Set up the shape of each matrix

There are two families (A and B) and three fruits (oranges, mangoes, bananas). Let the rows stand for the families and the columns stand for the fruits. Each week's data then fits neatly into a matrix with 2 rows and 3 columns, written as a 2 by 3 matrix. Keeping the rows and columns in the same order for both weeks is what makes the next step possible.

rows=families (A,B),columns=oranges, mangoes, bananas\text{rows} = \text{families } (A, B), \quad \text{columns} = \text{oranges, mangoes, bananas}

A matrix labelled 2 by 3 means 2 rows and 3 columns, in that order.

2

Write one matrix for each week

Record week 1 in its own 2 by 3 matrix and week 2 in another, using exactly the same layout. For example, the entry in row A, column oranges is 7 in week 1 and 6 in week 2, and row A, column mangoes is 15 in week 1 and 10 in week 2. This satisfies part (a): the information is now captured as two matrices of the same size.

W1=(715⋯⋯⋯⋯),W2=(610⋯⋯⋯⋯)W_1 = \begin{pmatrix} 7 & 15 & \cdots \\ \cdots & \cdots & \cdots \end{pmatrix}, \quad W_2 = \begin{pmatrix} 6 & 10 & \cdots \\ \cdots & \cdots & \cdots \end{pmatrix}

The narrator reads out only some of the entries aloud; the totals below confirm the rest.

3

Add the matrices position by position

To combine the two weeks you add the matrices. Because they are the same size, you simply add the numbers that sit in matching positions: same row, same column. So the two oranges entries for family A add to give that family's orange total, the two mangoes entries add to give its mango total, and so on across both rows.

T=W1+W2T = W_1 + W_2
7+6=13(oranges, family A)7 + 6 = 13 \quad (\text{oranges, family } A)
15+10=25(mangoes, family A)15 + 10 = 25 \quad (\text{mangoes, family } A)
4

Read off the total matrix

Carrying out every position-by-position addition gives a single 2 by 3 total matrix. Row A shows family A's totals and row B shows family B's totals, with the columns still in the order oranges, mangoes, bananas.

T=(13259233511)T = \begin{pmatrix} 13 & 25 & 9 \\ 23 & 35 & 11 \end{pmatrix}

Final Result

Family A bought 13 oranges, 25 mangoes, and 9 bananas in total. Family B bought 23 oranges, 35 mangoes, and 11 bananas in total, given by the total matrix with rows for the families and columns for the fruits.

Why this method works

Matrix addition works here because both weeks share the exact same structure — the same rows for families and the same columns for fruits. When two matrices have identical shape, adding matching positions just means adding the quantities that describe the same family and the same fruit, so each entry of the sum is a genuine two-week total. This is why keeping the rows and columns in a consistent order in part (a) is essential: it guarantees that position by position addition lines up like with like.

✓

Add the two known entries again to confirm the method: 7 + 6 = 13 and 15 + 10 = 25, matching the first row of the total matrix.