(a)

Consider upper triangular matrix, lower triangular matrix diagonal matrices are

Upper triangular matrix is

Determinant of upper triangular matrix is

.

.

Determinant of upper triangular matrix is .

Lower triangular matrix is

Determinant of lower triangular matrix is

.

Determinant of lower triangular matrix is .

Diagonal matrix is .

Determinant of diagonal matrix is

.

.

Determinant of diagonal matrix is .

(b)

Consider upper triangular matrix, lower triangular matrix and diagonal matrices are

Upper triangular matrix is .

Determinant of upper triangular matrix is .

.

Lower triangular matrix is .

Determinant of lower traingular matrix is .

.

Diagonal matrix is .

Determinant of diagonal matrix is .

.

.

(c)

Determinant of general  upper triangular matrix and lower triangular matrix and diagonal matrices are

The general matrix is 

Determinant of upper traingular matrix is .

.

Lower triangular matrix is

Determinant of lower traingular matrix is .

.

.

(d)

Inverse of matrix is

(e)

The general diagonal matrix is

The general inverse diagonal matrix is

The general inverse diagonal matrix is

(a) Determinant upper triangular matrix , lower triangular matrix and diagonal matrices are , and .

(b) Determinant upper triangular matrix, lower triangular matrix and diagonal matrices are ,and .

(c) Determinant of general  upper triangular matrix and lower triangular matrix and .

(d) Inverse of .

(e) The general inverse diagonal matrix is .