2. (a) (2 points) Let A be an n x n nonsingular lower triangular matrix with all of its nonzero entries on three diagonals of the matrix the main diagonal, the first sub- diagonal and the second sub-diagonal; that is, a1,1 a2,1 a22 , 1 a3.2 a3.3 4,2 44,3 44 an-1,p-3 an-1,n-2 an-1,-1 ann-2- an,n Note that A is nonsingular if and only if all of the entries on its main diagonal are nonzero, that each row of A contains at most 3 nonzero entries, and that entries on the first sub-diagonal and the second sub-diagonal may be nonzero or zero. For example, ifn-, then 3 0 0 0 00 -2 1.1 2.3 0 0 00 0 2.2 3-2.11 0 0 0 0 0 2 3.3 0 0 0 0 2.4 1-2.5 is such a lower triangular matrix Let b = [b1, b2.. .. , bn] denote a (column) vector with n entries. An n n system of linear equations Ar -b, where A is as described above, can be efficiently solved by forward substitution, which is similar to back-substitution but starts with the first equation. That is, the first equation can be used to solve for , the second equation can be used to solve for x2, the third equation for r, and so on. 2. (a) (2 points) Let A be an n x n nonsingular lower triangular matrix with all of its nonzero entries on three diagonals of the matrix the main diagonal, the first sub- diagonal and the second sub-diagonal; that is, a1,1 a2,1 a22 , 1 a3.2 a3.3 4,2 44,3 44 an-1,p-3 an-1,n-2 an-1,-1 ann-2- an,n Note that A is nonsingular if and only if all of the entries on its main diagonal are nonzero, that each row of A contains at most 3 nonzero entries, and that entries on the first sub-diagonal and the second sub-diagonal may be nonzero or zero. For example, ifn-, then 3 0 0 0 00 -2 1.1 2.3 0 0 00 0 2.2 3-2.11 0 0 0 0 0 2 3.3 0 0 0 0 2.4 1-2.5 is such a lower triangular matrix Let b = [b1, b2.. .. , bn] denote a (column) vector with n entries. An n n system of linear equations Ar -b, where A is as described above, can be efficiently solved by forward substitution, which is similar to back-substitution but starts with the first equation. That is, the first equation can be used to solve for , the second equation can be used to solve for x2, the third equation for r, and so on