# 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 $A$.

## 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*} Sponsored Links ### More from my site • Find Inverse Matrices Using Adjoint Matrices Let A be an n\times n matrix. The (i, j) cofactor C_{ij} of A is defined to be \[C_{ij}=(-1)^{ij}\det(M_{ij}), where $M_{ij}$ is the $(i,j)$ minor matrix obtained from $A$ removing the $i$-th row and $j$-th column. Then consider the $n\times n$ matrix […]
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##### 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...

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