Prove that an nxn upper triangular matrix has rank n if and only if there are no

Question

Prove that an nxn upper triangular matrix has rank n if and only if there are no zero elements on the diagonal.

Explanation / Answer

suppose the rank of the matrix nxn is r rxr has the determinant zero. by the properties of determinants, we follow that by application of elementary operations, the determinant is not changed. so, the determinant of the submatrix of order > rxr will be zero by considering the zeroeth row when multiplied with the respective minors. that means atleast one row of the determinant becomes entirely zero by the application of elementary operations on the given matrix. that means atleast one of the diagonal entries will be zero. ---------------------------------------------------------------------- conversely, suppose B is the upper triangular matrix in which atleast one of the diagonal entries is zero. by the properties of determinants, we know that the determinant of the matrix is nothing but the product of the diagonal entries when the given matrix A is changed into the upper triangular matrix with the help of elementary operations. so, the determinant of A = product of diagonal entries of B but one of the diagonal entries is zero in B. therefore, determinant of A = 0 and A is of order nxn. by definition of rank, we follow that rank of A

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