Quiz on observing same totals from row and column addition in arrays

📚 Quick Study Guide

• 🔢 Definition: An array exhibits equal row and column sums when the sum of all elements in each row equals the sum of all elements in each column.
• ➕ Calculation: To check, add up all numbers in each row and then add up all numbers in each column.
• ⚖️ Verification: If the total obtained from adding rows is the same as the total obtained from adding columns, the array fulfills the condition.
• 💡 Example: Consider a 2x2 array: [[1, 2], [2, 1]]. Row sums: 1+2=3, 2+1=3. Column sums: 1+2=3, 2+1=3.
• 📝 Formula: For an $m \times n$ array $A$, let $R_i$ be the sum of the $i$-th row and $C_j$ be the sum of the $j$-th column. The condition is satisfied if $\sum_{i=1}^{m} R_i = \sum_{j=1}^{n} C_j$.

Practice Quiz

1. Which of the following arrays has equal row and column sums?

   1. [[1, 2], [3, 4]]
   2. [[1, 2], [2, 1]]
   3. [[1, 3], [2, 4]]
   4. [[5, 5], [6, 6]]

2. Consider the array [[4, 5, 6], [7, 8, 9]]. What is the difference between the row sum and the column sum?

   1. 0
   2. 1
   3. 2
   4. 3

3. Which array does NOT exhibit equal row and column totals?

   1. [[1, 1, 1], [1, 1, 1], [1, 1, 1]]
   2. [[1, 2], [2, 1]]
   3. [[1, 0], [0, 1]]
   4. [[1, 2], [3, 4]]

4. For array [[a, b], [c, d]] to have equal row and column sums, which condition must be true?

   1. a + b = c + d
   2. a + c = b + d
   3. a + d = b + c
   4. a = b = c = d

5. Which of the following 3x3 arrays shows same totals from row and column addition?

   1. [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
   2. [[1, 2, 3], [2, 3, 1], [3, 1, 2]]
   3. [[1, 1, 1], [2, 2, 2], [3, 3, 3]]
   4. [[1, 0, 0], [0, 1, 0], [0, 0, 1]]

6. If an array has row sums equal to 10, what must be true for the array to have equal row and column sums?

   1. The column sums must also equal 10
   2. The column sums must equal 5
   3. The column sums must equal 20
   4. The array must be a square matrix

7. What is the sum of all elements in the array [[1, 2, 3], [1, 2, 3], [1, 2, 3]]? Does it have equal row and column sums?

   1. Sum is 9, and it does not have equal sums
   2. Sum is 18, and it has equal sums
   3. Sum is 18, and it does not have equal sums
   4. Sum is 9, and it has equal sums