In this square, all rows and columns have the same sum.Find the average of A𝐴, B𝐵, C𝐶 and D𝐷.

To find the average of $A, B, C,$ and $D$ in a square where all rows and columns have the same sum, we can follow these steps:

1. Break Down the Problem

1. Identify the structure of the square matrix and the properties of its elements.
2. Determine the relationships among $A, B, C,$ and $D$.

2. Relevant Concepts

• In a square matrix, each row and each column must sum to the same total value, known as the magic constant.
• The average of four numbers can be calculated as: $\text{Average} = \frac{A + B + C + D}{4}$

3. Analysis and Detail

Assuming the square is a $2 \times 2$ matrix: $\begin{bmatrix} A & B \\ C & D \end{bmatrix}$ The sums of the rows and columns are:

• Row 1: $A + B$
• Row 2: $C + D$
• Column 1: $A + C$
• Column 2: $B + D$

Given that all rows and columns have the same sum, we can equate these:

1. $A + B = C + D$
2. $A + C = B + D$

From these equations, relationships can be formed, allowing us to express one variable in terms of the others.

4. Verify and Summarize

To find the average: $\text{Average} = \frac{A + B + C + D}{4}$ The total sum $S$ of the matrix entries gives: $S = A + B + C + D$

Final Answer

The average of $A, B, C,$ and $D$ is: $\text{Average} = \frac{S}{4}$ Here, $S$ represents the combined total of $A, B, C,$ and $D$. If given specific values, substitute them in to compute the exact average.