1. Let A = 90 3 552 Find the characteristic polynomial of A and guess one...

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1. Let A = 90 3 552 Find the characteristic polynomial of A and guess one of its roots by looking at integer divisors of the constant term. Find an eigenvector corresponding to this eigenvalue, and find the remaining eigenvalues of A. 2. Let A be an n x n matrix. (a) Prove that A is singular if and only if A = 0 is an eigenvalue of A. (A square matrix is said to be singular if it is not invertible.) (b) Suppose that A is invertible, and v is an eigenvector of A, corresponding to some eigenvalue = R. Show that v is an eigenvector of A-1, corresponding to the eigenvalue = 1/X. 3*. (This question will not be marked. It is included here with the purpose to provide more examples.) Suppose that A M(R) and v,...,V R are eigenvectors of A corresponding to the same eigenvalue A E R. Prove that if u is a linear combination of vectors V,..., V, then Au = Au. Conclude that any non-zero vector u span{v,..., V} is an eigenvector of A corresponding to the eigenvalue A. 4. Suppose that P is an n x n orthogonal matrix. (a) Prove that the columns of P form an orthonormal set in R". (b) Show that either det(P) = 1 or det(P) = -1. 1. Let A = 90 3 552 Find the characteristic polynomial of A and guess one of its roots by looking at integer divisors of the constant term. Find an eigenvector corresponding to this eigenvalue, and find the remaining eigenvalues of A. 2. Let A be an n x n matrix. (a) Prove that A is singular if and only if A = 0 is an eigenvalue of A. (A square matrix is said to be singular if it is not invertible.) (b) Suppose that A is invertible, and v is an eigenvector of A, corresponding to some eigenvalue = R. Show that v is an eigenvector of A-1, corresponding to the eigenvalue = 1/X. 3*. (This question will not be marked. It is included here with the purpose to provide more examples.) Suppose that A M(R) and v,...,V R are eigenvectors of A corresponding to the same eigenvalue A E R. Prove that if u is a linear combination of vectors V,..., V, then Au = Au. Conclude that any non-zero vector u span{v,..., V} is an eigenvector of A corresponding to the eigenvalue A. 4. Suppose that P is an n x n orthogonal matrix. (a) Prove that the columns of P form an orthonormal set in R". (b) Show that either det(P) = 1 or det(P) = -1.