The product of matrix and its transpose matrix is zero matrix, and it is proved that the original matrix is zero matrix

The product of matrix and its transpose matrix is zero matrix, and it is proved that the original matrix is zero matrix

Let's just expand the matrix and write it as
A=(a11 a12…… a1n
a21 a22…… a2n
………………
an1 an2…… ann）
Then write a 'directly, multiply it directly, and focus on the elements on the main diagonal

What is the relationship between the product of a matrix equal to the sum of zeros and ranks
Let AB = 0, a be mxn, B be NXS matrix, then the column vectors of B are AX = 0, so r (b)

There are N-R (a) vectors in the fundamental solution system of homogeneous linear equations AX = 0
Each column of B, as the solution vector of AX = 0, can be linearly expressed by the basic solution system,
So r (b) ≤ N-R (a)

If the rank of n * n matrix A is n, then the rank of the adjoint matrix of a is n; if R (a) = n-1, then it is 1; if R (a) = n-1, then it is 1