Wednesday, February 2, 2011 

On Wednesday, mathematicians and algebra students are celebrating “Linear Algebra Day”, as it is the second day of the second month, and most students learning linear algebra and matrices for the first time start with 2×2 matrices since they are the most basic and easiest to work with, besides 1×1, which is trivial.

The determinant of a 2×2 matrix is calculated by subtracting the diagonal products: det(M) = ad − bc.

For example, for the matrix

5; 7

6; 10

The determinant of the matrix is 5 × 10 − 7 × 6 = 50 − 42 = 8.

Although 2×2 matrices have determinants, non-square matrices do not, so this property does not apply to any other day of this month. Tomorrow is 2/03, and 2×3 matrices do not have determinants. Neither do 2×4 matrices, so Friday, February 4th doesn’t work, either. Now, it’s easy to see that the only dates of the year that correspond to square matrices are the twelve dates for which the month and the day have the same number.

• January 1
• February 2
• March 3
• April 4
• May 5
• June 6
• July 7
• August 8
• September 9
• October 10
• November 11
• December 12

The next day of this kind is Thursday, March 3, 2011. Note that 3×3 matrices are square, so they have determinants and inverses, just like 2×2 matrices do.

If M is a square matrix, and x and y are vectors, then the system Mx = y has a unique solution if and only if M is of full rank. If the matrix M has less than full rank, this system may have either no solutions or infinitely many solutions. There also exist vectors v ≠ 0 such that Mv = 0. The set (or space) of vectors that satisfy that equation, sometimes called the homogeneous equation, is called the kernel or the null space. For full-rank matrices, the null space consists of only one vector: the zero vector.

• “[ ]” —  e.g. December 31, 1999
• “[ ]” —