It turns out that skew-symmetric nilpotent 3x3 matrices satisfy the equation a^2+b^2+c^2=0, where a,b, and c are the off diagonal elements. If you write two such matrices, and force them to commute, the equations show that (a,b,c) and (a',b',c') must be scalar multiples. I'm hoping to find a more elegant way, because the same method for 4x4 Determinant of a 4×4 matrix is a unique number that is also calculated using a particular formula. If a matrix order is in n x n, then it is a square matrix. So, here 4×4 is a square matrix that has four rows and four columns. If A is a square matrix then the determinant of the matrix A is represented as |A|. A cofactor corresponds to the minor for a certain entry of the matrix's determinant. To find the cofactor of a certain entry in that determinant, follow these steps: Take the values of i and j from the subscript of the minor, Mi,j, and add them. Take the value of i + j and put it, as a power, on −1; in other words, evaluate (−1)i+j. The other problems can be found from the links below. Find All the Eigenvalues of 4 by 4 Matrix (This page) Find a Basis of the Eigenspace Corresponding to a Given Eigenvalue. Diagonalize a 2 by 2 Matrix if Diagonalizable. Find an Orthonormal Basis of the Range of a Linear Transformation. The inverse of matrix A can be computed using the inverse of matrix formula, A -1 = (adj A)/ (det A). i.e., by dividing the adjoint of a matrix by the determinant of the matrix. The inverse of a matrix can be calculated by following the given steps: Step 1: Calculate the minors of all elements of A. tions of this determinant of 4 4 are reduced, this book does not give examples with larger matrixs. The proof of this expression appears in a 6th unit theorem of the book: "Notes on the combinatorial fundamentals of algebra", by Darij Grinberg [2], and another proof in: "Matrix Canonical Forms ", by S. Gill Williamson on page 49 [8] .

determinant of a 4x4 matrix example