Please i need help with this problem. determine whether the given set of matrice
ID: 2260868 • Letter: P
Question
Please i need help with this problem.
determine whether the given set of matrices under the specified operation, matrix addition is a group Recall that a diagonal matrix square matrix whose only nonzero entries the main diagonal, from the upper left to the lower right corner. An upper-triangular matrix is a square matrix with only zero entries below the main diagonal. Associated with each n X n matrix A is a number called the determinant of A, denoted by det(A). If A and B are both n x n matrices, then det(AB)=det(A)det(R). Also, det(1n)=1 and A is invertible if and only if det(A)not equal 0.
All n x n upper-triangular matrices under matrix addition.
Explanation / Answer
(i) It is fairly straightforward (although tedious to write) to see that the sum of two upper triangular matrices again is upper triangular.
(ii) The identity matrix is upper triangular, because it has only 1's as its diagonal terms.
(iii) Via (i), the inverse matrix will also be upper triangular.
Another way to see this is using A^(-1) = (1/det A) * A*, where A* is the matrix of signed cofactors. It is easy to check that A* is upper triangular.
Moreover, det(A^(-1)) = 1/det(A) = 1/1 = 1.
(iv) Matrix addition is associative (A+B=B+A). Therefore any subset of matrices will also enjoy associativity.