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Mathematics Math21b Spring 2008

Linear Algebra and Differential Equations

Exhibit: latin squares

Course Head: Oliver Knill

Office: SciCtr 434

Email: knill@math.harvard.edu

## Latin squaresA nxn matrix is called a latin square if the numbers 1,...,n occur exactly once in each row and exactly once in each column of the matrix. A simple example isA = | 1 2 | | 2 1 |An example of a 3x3 latin square, you have encountered in the first midterm. Here is an example of a 5x5 latin square: A = | 1 2 3 4 5 | | 2 3 5 1 4 | | 3 5 4 2 1 | | 4 1 2 5 3 | | 5 4 1 3 2 | |

A 9x9 matrix is a Suduku square,
if it is a latin square and if additionally in
each of the 9 3x3 submatrices all numbers 1..,9 occur exactly once too.
Here is an example:
A = | 3 4 2 9 7 8 1 5 6 | | 6 9 5 2 4 1 3 7 8 | | 1 7 8 6 3 5 2 4 9 | | 7 6 3 4 9 2 5 8 1 | | 8 1 9 3 5 7 4 6 2 | | 2 5 4 8 1 6 7 9 3 | | 9 3 7 1 6 4 8 2 5 | | 4 8 1 5 2 9 6 3 7 | | 5 2 6 7 8 3 9 1 4 | |

A nxn matrix is called a Magic square if it contains all
integers 1,...,n^{2} exactly once and each row each column
and each diagonal column has the property that the sum is constant.
An example:
| 4 9 2 | | 3 5 7 | | 8 1 6 |Note that this square appears in the center of the above Suduku matrix which has been found by Paul Muljadi. A = | 16 3 2 13 | | 5 10 11 8 | | 9 6 7 12 | | 4 15 14 1 | |

- Which of the three classes of matrices has the property that its transpose A
^{T}has the same property too? - Can you find an eigenvalue and eigenvector in each of the above three classes of matrices?
- Can you see why every Suduku matrix has a nontrivial kernel and is therefore not invertible?

Please send questions and comments to math21b@fas.harvard.edu

Math21b | Oliver Knill | Spring 2008 |
Department of Mathematics |
Faculty of Art and Sciences |
Harvard University