Question
la question 10
(b) Triang of a lower triangular matrix is upper triangular, and the transpose of an trian (c) A triangular matrix is i 10) lheorern: (a) The trans (a) The transplar mia trix is rSs1 rtland only .1atrix dar. uperduct of lan pper t er triangt of lower thr triat aulaiang ispose pose of a l The ar ma ofTinerse oriatrix is tnvertible if and only if its diagonal entries are all nonzer of an invertible upper of lower ces is roductatrix is lower triangular matricowr triangular matrices is lower triangular, and the product of upper d) The ul"pper triangular. atrix is invertible if and only if its diagonal entries are all nonzero. ertibef an invertible lower triangular matrix is lower triangular, and the inverse epper triangular matrix is upper triangular. Consider the upper triangular matrices 1 3 -1 3-2 2 A 0 2 4 and B0 0-1 0 0 1 0 0 5 Verifies that (a) the transpose of an A is lower triangular. (b)the product of A and B is upper triangular. by finding the inverse matrix, show that A is invertible but B is not. (d) show that A-1 is an upper trianglar matrix. (c)
Explanation / Answer
a) A^T matrix
Since the A^T has three corners zero in the upper side, hence the matrix A^T is lower triangluar matrix
b) The product of A and B will be upper triangular matrix
The product of matrix AB will be an upper triangular
c) The determinant of A is
det(A) = 1(10) + 3(0-0) -1(0) = 10
The determinant of B is
3(0+0) -2(0-0) + 2(0-0) = 0
Since the determinant of B is zero, hence the inverse of B doesn't exists, therefore A is invertible and B is not
d)
The inverse of the matrix A will be
A^(-1) = 1/|A| adj(A)
1 0 0 3 2 0 -1 4 5
