2) Let A be an nxn matrix with eigenvalue of multiplicity n. Show that Ais diagonalizable if and only if A = al.
2) Let A be an nxn matrix with eigenvalue a of multiplicity n. Show that A is diagonalizable if and only if A= 21.
Let A be a nxn matrix and let B be the matrix composed of co-factors of the matrix A. Show that if rk(A) = n - 1 then rk(B) = 1. Explain.
6. Let A be a nxn matrix and let B be the matrix composed of co-factors of the matrix A. Show that if rk(A) = n - 1 then rk(B) = 1. Explain.
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
21 22 23 24 If the matrix Al is diagonalizable, then the matrix A must be diagonalizable as well. The determinant of a matrix is the product of its eigenvalues, counted with their algebraic multiplicities. All lower triangular matrices are diagonalizable. If two nxn matrices A and B are diagonalizable, then AB must be diagonalizable as well. If an invertible matrix is diagonalizable, then A-1 must be diagonalizable as well. 25
Let A be an m*n matrix. Prove that AA(transpose) is orthogonally diagonalizable.
Let AA be an m×nm×n matrix. Prove that AATAAT is orthogonally diagonalizable.
DETAILS LARLINALG8 7.2.050. Show that the matrix is not diagonalizable. [ ] : 0 The matrix is not diagonalizable because it only has one linearly independent eigenvector. The matrix is not diagonalizable because it only has one distinct eigenvalue. The matrix is not diagonalizable because [*] is not an eigenvector. The matrix is not diagonalizable because k is not an eigenvalue.
SOLVE BOTH 4 and 5!!
4. Let A and B be two nxn matrices. Suppose that AB is invertible. Show that the system Ar 0 has only the trivial solution 5. Given that B and D are invertible matrices of orders n and p respectively, and A = Find A by writing A as a suitably partitioned matrix