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GeezerVison is showing mathworld.wolfram.com — Square Matrix -- from Wolfram MathWorld

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Square Matrix

A square matrix is a matrix for which horizontal and vertical dimensions are the same (i.e., an n×n matrix).

A matrix m may be tested to determine if it is square in Wolfram Language using SquareMatrixQ[m].

Let n be a positive integer. Consider the numbers of n×n matrices containing each of n^2 distinct symbols exactly once. The number of distinct matrices modulo rotations and reflections for n=1, 2, ... are given by 1, 3, 45360, ... (OEIS A086829).

Consider an n×n matrix consisting of the integers 1 to n^2 arranged in any order. Then the maximal determinants possible for n=1, 2, ... are 1, 10, 412, 40800, 6839492, ... (OEIS A085000).

Consider an n×n matrix with single copies of the integers 1, 2, ..., d and the rest of the elements zero. Then the triangle of n×n matrices with d=0, 1, ..., n^2 distinct nonzero entries that are rotationally and reflectively distinct is 1, 1; 1, 1, 2, 3, 3; 1, 3, 12, 66, 378, 1890, 7560, 22680, 45360, 45360; ... (OEIS A087074).

See also

Array, Magic Square, Matrix, Rectangular Matrix, Square Array

Explore with Wolfram|Alpha

WolframAlpha

More things to try:

  • eigenvalues of matrix [(2,2,1),(1,3,1),(1,2,2)]
  • eigenvalues of square matrix {{2,2,1},{1,3,1},{1,2,2}}
  • find the eigenvalues of matrix [-1, 2] [-5, 1]

References

Sloane, N. J. A. Sequences A085000, A086829, and A087074 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Square Matrix

Cite this as:

Weisstein, Eric W. "Square Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SquareMatrix.html

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