Question

If \( A = \begin{bmatrix} a & b & 7 \\ c & 8 & 3 \\ d & e & f \end{bmatrix} \) is a matrix such that the sum of all three elements along any row, column, or diagonal are equal to each other, then the value of the determinant of \( A \) is?

Hint:

When working with matrices where row, column, or diagonal sums are equal, check if the matrix can be transformed into a square matrix or use determinant properties specific to such structured matrices.

Answer 1

Since the sum of the elements along any row, column, or diagonal is equal, this is a magic square. Let the magic constant be \( S \). The sum of all elements in the matrix is \( a + b + 7 + c + 8 + 3 + d + e + f \). Since each row sums to \( S \), the sum of all elements is also \( 3S \). The sum of integers from 1 to 9 is \( \frac{9(10)}{2} = 45 \). Thus, \( 3S = 45 \), which implies \( S = 15 \). Now, we can find the missing elements:

• Row 2: \( c + 8 + 3 = 15 \implies c = 4 \)
• Column 3: \( 7 + 3 + f = 15 \implies f = 5 \)
• Diagonal 1: \( a + 8 + 5 = 15 \implies a = 2 \)
• Row 1: \( 2 + b + 7 = 15 \implies b = 6 \)
• Column 2: \( 6 + 8 + e = 15 \implies e = 1 \)
• Row 3: \( d + 1 + 5 = 15 \implies d = 9 \)

Therefore, the matrix \( A \) is: $$ A = \begin{bmatrix} 2 & 6 & 7 \\ 4 & 8 & 3 \\ 9 & 1 & 5 \end{bmatrix} $$ Now, we calculate the determinant of \( A \): \[ \det(A) = 2(8 \cdot 5 - 3 \cdot 1) - 6(4 \cdot 5 - 3 \cdot 9) + 7(4 \cdot 1 - 8 \cdot 9) \] \[ = 2(40 - 3) - 6(20 - 27) + 7(4 - 72) \] \[ = 2(37) - 6(-7) + 7(-68) \] \[ = 74 + 42 - 476 \] \[ = 116 - 476 \] \[ = -360 \] Answer: The determinant of \( A \) is \( -360 \).