1. The problem is to find the missing numbers in the last row of the matrix: $$\begin{matrix} 38 & 71 & 78 & 73 & 86 \\ 85 & 33 & 17 & 42 & 99 \\ 99 & 22 & 47 & 39 & 31 \\ 87 & 22 & 21 & ?? & ?? \end{matrix}$$ 2. To solve this, we need to identify a pattern or rule that relates the rows or columns. 3. Let's check if the sum of each row is consistent: Row 1 sum: $38 + 71 + 78 + 73 + 86 = 346$ Row 2 sum: $85 + 33 + 17 + 42 + 99 = 276$ Row 3 sum: $99 + 22 + 47 + 39 + 31 = 238$ Row 4 sum so far: $87 + 22 + 21 = 130 + ?? + ??$ No clear pattern in sums. 4. Let's check the sum of each column: Column 1: $38 + 85 + 99 + 87 = 309$ Column 2: $71 + 33 + 22 + 22 = 148$ Column 3: $78 + 17 + 47 + 21 = 163$ Column 4: $73 + 42 + 39 + ?? = 154 + ??$ Column 5: $86 + 99 + 31 + ?? = 216 + ??$ No obvious pattern in columns either. 5. Let's check if the last two numbers in the last row can be found by subtracting the third number from the first two numbers in the same row: $87 - 22 = 65$ $22 - 21 = 1$ But these do not match the missing numbers. 6. Another approach is to check if the missing numbers are the differences between the sums of the previous rows: Sum of row 1: 346 Sum of row 2: 276 Sum of row 3: 238 Difference between row 3 and row 4 sums: $238 - (87 + 22 + 21 + x + y) = 0$ Let $x$ and $y$ be the missing numbers. Sum of known numbers in row 4: $87 + 22 + 21 = 130$ So $130 + x + y = 238$ Therefore, $x + y = 108$ 7. Without additional information, we can only say the sum of the missing numbers is 108. Final answer: The missing numbers in the last row add up to 108, but cannot be determined uniquely with the given data.