6.3 (Adjugate matrix) a) Let A E GLn(K). Use Cramer's rule to show that A-1 =...
3. [2+2pt] Let n > 2. Consider a matrix A E Rnxn for which every leading principal submatrix of order less than n is non-singular. (a) Show that A can be factored in the form A = LDU, where Le Rnxn is unit lower triangular, D e Rnxn is diagonal and U E Rnxn is unit upper triangular. (b) If the factorization A = LU is known, where L is unit lower triangular and U is upper triangular, show how...
7. Let A [aij] be an n x n invertible tridiagonal matrix, that is aij= 0 if |i - j > 1. Compute the number of operations needed to solve the system Ax b by Gauss elimination without partial pivoting. (10 marks) 7. Let A [aij] be an n x n invertible tridiagonal matrix, that is aij= 0 if |i - j > 1. Compute the number of operations needed to solve the system Ax b by Gauss elimination without...
(7 marks) Let the distinct eigenvalues of a square matrix A be denoted by 11, ..., lk. Suppose the corresponding algebraic multiplicities are mi, ..., Mk and that A is similar to an upper-triangular matrix. Show that k k tr(A) midi and det(A) = II (4;)mi i=1 i=1
(12) (7 marks) Let the distinct eigenvalues of a square matrix A be denoted by 11, ..., dk. Suppose the corresponding algebraic multiplicities are m1, ..., mk and that A is similar to an upper-triangular matrix. Show that k tr(A) = midi and det(A) = II (1;)" i=1 i=1
Let A-(Aij)i iJSn є {0,1)"xn denote the symmetric adjacency matrix of an undi- rected graph. For iメj, we have Aij = 1 if entity i and j are connected in a network and 0 otherwise: A 0, i-1,..., n. The stochastic block model (SBM) postulates where is a full rank symmetric K x K connectivity matrix with entries in [0, 1]. a) Consider the matrix P-M MT, where M {0,1)"xK denotes the community k-1,... , K. Show that under (1),...
Let A be an n×n matrix. Mark each statement as true or false. Justify each answer. a. An n×n determinant is defined by determinants of (n−1)×(n−1) submatrices. b. The (i,j)-cofactor of a matrix A is the matrix obtained by deleting from A its I’th row and j’th column. a. Choose the correct answer below. A. The statement is false. Although determinants of (n−1)×(n−1)submatrices can be used to find n×n determinants,they are not involved in the definition of n×n determinants. B....
4 (1) Find a matrix A „such that (A - 41)-1 3 1 (2) Let A be 3x3 matrix with 4 = 4 Find : (a) det(( 3 A)?(2 A)-') (b) det( 2 A-' + 3 adj (A)) (3)Find the values of a that makes the system has (a) unique solution (b) No Solution. 3 A 7 (4)Find the rank of a matrix 17 0 1 2 (5)Suppose that I : R3 → R2 „such that 2 T (e.) =...
3. Let U E Rnxn be an orthogonal matrix, i.c., UTU = UUT-1. Show that for any vector x E Rn LXTL we have |lU 2 2. Thus the 2-norm of a vector does not change when it is multiplied by an orthogonal matrix. 3. Let U E Rnxn be an orthogonal matrix, i.c., UTU = UUT-1. Show that for any vector x E Rn LXTL we have |lU 2 2. Thus the 2-norm of a vector does not change...
U PETEKU UEU 1. Let B = Find x when matrix B has no inverse. (4) -4 2 - X J 2. Find the value (s) of a for which the system of equations below has infinitely many solutions. (a - 3x + y = 0 x + (a-3)y=0 3. Let A be a symmetric matrix. Show that A2 is also symmetric. 4. Find matrix A for which A-* = [ 2 ar flimaaranations and use Cramer's rule to find