-12 Q2. Find the matrix A knowing that (12 +2A) = . Where I is an...
Show that if A is a square matrix that satisfies the equation A2 - 2A + I = O, then A = 2I - A The equation A - 2A I = O implies that It follows that Notice ---Select--- ---Select--- the last equation means that multiplied with A is the identity, which is what we wanted to prove. -Select--
Question B
7. (a) Let -1 0 0 (i) Find a unitary matrix U such that M-UDU where D is a diagonal matrix. 10 marks] (i) Compute the Frobenius norm of M, i.e., where (A, B) = trace(B·A). [4 marks] 3 marks] (iii) What is NM-illp? (b) Let H be an n × n complex matrix (6) What does it mean to say that H is positive semidefinite. (il) Show that H is positive semidefinite and Hermitian if and only...
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
Let A be a square matrix. Prove that if A2 = A, then I - 2A is the inverse of I - 2A.
Please show full workings only answer if you know how.
(5) Consider the 3 x 3 matrix A - I - avv7 where a e R. I is the identity matrix and v the vector 1S 2 (a) Determine the eigenvalues and eigenvectors of A (b) Hence find a matrix which diagonalises A. (c) For which a is the matrix A singular? (d) For which α is the matrix A orthogonal ?
(5) Consider the 3 x 3 matrix A...
An invertible square matrix A satisfies A^3 +3A^2 −25A+21I = O, where I and O are the identity and zero matrices, respectively. Find the inverse of A^2
Consider the 3 x 3 matrix A-1-ovvT where a R, 1 is the identity matrix and v the vector (a) Determine the eigenvalues and eigenvectors of A (b) Hence find a matrix which diagonalises A. (c) For which a is the matrix A singular? (d) For which α is the matrix A orthogonal ?
Consider the 3 x 3 matrix A-1-ovvT where a R, 1 is the identity matrix and v the vector (a) Determine the eigenvalues and eigenvectors of...
Find
the eigenvalues and associated eigenvectors of the matrix
Q2: Find the eigenvalues and associated eigenvectors of the matrix 7 0 - 3 A = - 9 2 3 18 0 - 8
Given the matrix A [1 7 L3 6 91 5 2. 4 8] (a) Find the inverse of the matrix A clearly showing all the steps leading to the inverse matrix. (b) Show clearly using matrix multiplication that AA-1 = I and A-1A = I, where I is the identity matrix.
(b) Let T E L(R2) where T(a,b) = (-2a + 36, -10a + 9b). Find the eigenvalues of T and an ordered basis B where (T]2 is a diagonal matrix.