Then it has (not have!) a unique solution.
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First, You have to reduce the matrix to echelon form . The number of nonzero rows in the reduced echelon form matrix (number of linearly independent rows) indicates the rank of the matrix. Go to any search engine and type "Rank of a matrix, Cliffnotes" for an example.
Yes it is. It is an ordinal variable ( which means it is meaningful ) because rank has an order and it is meaningful to rank the soldiers.
Rank of a matrix is used to find consistency of linear system of equations.As we know most of the engineering problems land up with the problem of finding solution of a linear system of equations ,at that point rank of matrix is useful.
The fact that the matrix does not have an inverse does not necessarily mean that none of the variables can be found.
Hermitian matrix (please note spelling): a square matrix with complex elements that is equal to its conjugate transpose.