Compute the Determinant of a Magic Square

Linear Algebra Problems and Solutions

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 $A$.

 
LoadingAdd to solve later

Sponsored Links

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*}


LoadingAdd to solve later

Sponsored Links

More from my site

  • Find Inverse Matrices Using Adjoint MatricesFind 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 […]
  • Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$ Determine whether there exists a nonsingular matrix $A$ if \[A^4=ABA^2+2A^3,\] where $B$ is the following matrix. \[B=\begin{bmatrix} -1 & 1 & -1 \\ 0 &-1 &0 \\ 2 & 1 & -4 \end{bmatrix}.\] If such a nonsingular matrix $A$ exists, find the inverse […]
  • Find All Values of $x$ such that the Matrix is InvertibleFind All Values of $x$ such that the Matrix is Invertible Given any constants $a,b,c$ where $a\neq 0$, find all values of $x$ such that the matrix $A$ is invertible if \[ A= \begin{bmatrix} 1 & 0 & c \\ 0 & a & -b \\ -1/a & x & x^{2} \end{bmatrix} . \]   Solution. We know that $A$ is invertible precisely when […]
  • How to Use the Cayley-Hamilton Theorem to Find the Inverse MatrixHow to Use the Cayley-Hamilton Theorem to Find the Inverse Matrix Find the inverse matrix of the $3\times 3$ matrix \[A=\begin{bmatrix} 7 & 2 & -2 \\ -6 &-1 &2 \\ 6 & 2 & -1 \end{bmatrix}\] using the Cayley-Hamilton theorem.   Solution. To apply the Cayley-Hamilton theorem, we first determine the characteristic […]
  • Find the Inverse Matrix Using the Cayley-Hamilton TheoremFind the Inverse Matrix Using the Cayley-Hamilton Theorem Find the inverse matrix of the matrix \[A=\begin{bmatrix} 1 & 1 & 2 \\ 9 &2 &0 \\ 5 & 0 & 3 \end{bmatrix}\] using the Cayley–Hamilton theorem.   Solution. To use the Cayley-Hamilton theorem, we first compute the characteristic polynomial $p(t)$ of […]
  • Compute Determinant of a Matrix Using Linearly Independent VectorsCompute Determinant of a Matrix Using Linearly Independent Vectors Let $A$ be a $3 \times 3$ matrix. Let $\mathbf{x}, \mathbf{y}, \mathbf{z}$ are linearly independent $3$-dimensional vectors. Suppose that we have \[A\mathbf{x}=\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, A\mathbf{y}=\begin{bmatrix} 0 \\ 1 \\ 0 […]
  • Quiz 11. Find Eigenvalues and Eigenvectors/ Properties of DeterminantsQuiz 11. Find Eigenvalues and Eigenvectors/ Properties of Determinants (a) Find all the eigenvalues and eigenvectors of the matrix \[A=\begin{bmatrix} 3 & -2\\ 6& -4 \end{bmatrix}.\] (b) Let \[A=\begin{bmatrix} 1 & 0 & 3 \\ 4 &5 &6 \\ 7 & 0 & 9 \end{bmatrix} \text{ and } B=\begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 &0 […]
  • Use Cramer’s Rule to Solve a $2\times 2$ System of Linear EquationsUse Cramer’s Rule to Solve a $2\times 2$ System of Linear Equations Use Cramer's rule to solve the system of linear equations \begin{align*} 3x_1-2x_2&=5\\ 7x_1+4x_2&=-1. \end{align*}   Solution. Let \[A=[A_1, A_2]=\begin{bmatrix} 3 & -2\\ 7& 4 \end{bmatrix},\] be the coefficient matrix of the system, where $A_1, A_2$ […]

You may also like...

Leave a Reply

Your email address will not be published. Required fields are marked *

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

More in Linear Algebra
Linear Transformation problems and solutions
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