Linear Algebra

Compute the Determinant of a Magic Square

Problem 718

Let
\[
A=
\begin{bmatrix}
8 & 1 & 6 \\
3 & 5 & 7 \\
4 & 9 & 2
\end{bmatrix}
.
\] Notice that $A$ contains every integer from $1$ to $9$ and that the sums of each row, column, and diagonal of $A$ are equal. Such a grid is sometimes called a magic square.

Compute the determinant of $A$.

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Solution.

We compute using the first row cofactor expansion
\begin{align*}
\det(A)
&=
\begin{vmatrix}
8 & 1 & 6 \\
3 & 5 & 7 \\
4 & 9 & 2
\end{vmatrix}\\[6pt] &=
8
\begin{vmatrix}
5 & 7 \\ 9 & 2
\end{vmatrix}
-1
\begin{vmatrix}
3 & 7 \\ 4 & 2
\end{vmatrix}
+6
\begin{vmatrix}
3 & 5 \\ 4 & 9
\end{vmatrix}
\\[6pt] &=
8(10-63)-(6-28)+6(27-20)\\[6pt] &=
8(-53)-(-22)+6(7)
\\[6pt] &=
-424+22+42\\[6pt] &=
-360.
\end{align*}

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