# 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 $A$. Add to solve later

## 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*} Add to solve later

### More from my site

#### You may also like...

This site uses Akismet to reduce spam. Learn how your comment data is processed.

###### More in Linear Algebra ##### Are These Linear Transformations?

Define two functions $T:\R^{2}\to\R^{2}$ and $S:\R^{2}\to\R^{2}$ by \[ T\left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} 2x+y \\ 0...

Close