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What is it when the first nonzero entry in each row of a matrix is one?
Nothing particular if all the 1s are in the first column, for example. You could have an echelon matrix, but with the information given, it is hard to tell.
If a matrix has only one element is it a square matrix?
Yes.
How do find matrix inverse of 4cross 4matrix?
Next to your 4x4 matrix, place the 4x4 identity matrix on the right and adjoined to the one you want to invert. Now you can use row operations and change your original matrix on the left to a 4x4 identity matrix. Each time you do a row operation, make sure you do the same thing to the rows of the original identity matrix. You end up with the identity now on the left and the inverse on the right. You can also calculate the inverse using the adjoint. The adjoint matrix is computed by taking the transpose of a matrix where each element is cofactor of the corresponding element in the original matrix. You find the cofactor t of the matrix created by taking the original matrix and removing the row and column for the element you are calculating the cofactor of. The signs of the cofactors alternate, just as when computing the determinant
Define sparse matrix?
A matrix that have one or more elements with value zero.
How do you show a variance matrix is nonnegative definite?
I assume you mean covariance matrix and I assume that you are familiar with the definition:C = E[(X-u)(X-u)T]where X is a random vector and u = E(X) is the meanThe definition of non-negative definite is:xTCx ≥ 0 for any vector x Є RSo is xTE[(X-u)(X-u)T]x ≥ 0?Then, from one of the covariance properties:E[(xT(X-u))((X-u)Tx)] = E[xT([(X-u)(X-u)T]x)] = E[((xTI)x)] = E[xTx]Finally, since we've already defined x to have only real values, xTx is therefore non-negative definite by definition.
