(1) If A is nx n then A+(41)+ = 1. (11) If A is an nxn matrix, then tr(CA) = (-1 tr(:4). (111) If A A is a symmetric matrix, then A must be a square matrix. Determine which of the above statements are True (1) or False (2).
Which of the following is an INCORRECT statement: 1) MATLAB command eye(n) makes an nxn identity matrix 2) Identity matrix is square matrix with ones on main diagonal and zeros elsewhere 0 3) Matrix multiplication on any array with the identity matrix changes the array 4) Matrix B is the inverse of matrix A if matrix product of A and B is the identity matrix
8] E! k=0 for all 2, prove the identity (P-Y = ee-
A scalar matrix is simply a matrix of the form XI, where I is the nxn identity matrix. (a) Prove that if A is similar 1 to \I, then in fact A= \I. (b) Show that a diagonalizable matrix having only one eigenvalue is a scalar matrix. 1 100 100 (c) Prove that o 100 is not diagonalizable. 0 0 1 1
Prove this identity
ssume A and B are invertible nxn matrices and k is a scalar. Prove the following. a.) If A is invertible, then 14-1 (1/(|4). (AB),I=1시1
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Let B and C be similar nxn matrices. Prove that the matrices given by: I +5B - 2B4 and I +5C - 204 are similar. (6 pts)
3) Let N- 11,2,3,... and Nx N -(m,n) | m,n E N. Consider f NxN-N given by f(1,2)-3 | f(2,2)-6 | fa, 21-12 f (1,3)-5 f (2,3)10f (3,3)- 20 f (1,4) 7 f (2,4) 14 f (3,4) 28 2m-i (2n-1). Show, that f is one-to-one and In general "f(m, n) onto.
3) Let N- 11,2,3,... and Nx N -(m,n) | m,n E N. Consider f NxN-N given by f(1,2)-3 | f(2,2)-6 | fa, 21-12 f (1,3)-5 f (2,3)10f (3,3)- 20...
*Prove that the orthogonal systems {sin nx)., and {ï, cos nxml are both complete on [0, T] 3.
Let f(3) = 1 (a) Prove {f} 1 + nx converges to 0 pointwise on (-0,00). (b) Prove or disprove {n} , converges to 0 uniformly on (-0, 0);