Math 21b: Practice questions for second midterm1. Let A be the matrix1 2 31 0 10 2 2,and let~b1,~b2be the vectors~b1=−133,~b2=431.(a) Compute det(A). What does this tell you about solutions of the linear system A~x =~bfor arbitrary~b?(b) Does A~x =~b1have an exact solution?(c) Does A~x =~b2have an exact solution?(d) Find an orthonormal basis for im(A).(e) Find a least-squares solution of A~x =~b1.(f) Find a least-squares solution of A~x =~b2.2. (a) [1 pt.] For a square matrix A, define the characteristic polynomial fA(λ), the algebraicmultiplicity of an eigenvalue, and the geometric multiplicity of an eigenvalue.(b) [2 pts.] For any scalar a let A be the matrix1 1 1−1 a 10 0 1.Compute the characteristic polynomial fA(λ).(c) [4 pts.] For which value(s) of a does A have an eigenvalue λ of algebraic multiplicity atleast 2? What is that λ in each case?(d) [2 pts.] For each of the cases you found in (c), what is the geometric multiplicity of λ?(e) [1 pt.] For each of the cases you found in (c), is A diagonalizable?3. Let A be the matrix1 43 2.(a) Compute the eigenvalues of A.(b) Find an eigenbasis for A.(c) Find a matrix S such that D = S−1AS is diagonal. What is D?(d) Give a formula for An. Check your work by verifying that the cases n = 1 and n = 2 ofyour formula agree with A and A2.1True or False?• If ~x and ~y are any vectors in Rnthen k~x + ~yk2+ k~x − ~yk2= 2(k~xk2+ k~yk2).• If A is a matrix and~b is a vector in ker ATthen~0 is a least-squares solution of A~x =~b.• There exists a subspace V of R3such that dim V⊥= dim V .• If a matrix A has tr(A) = 0 then A is not invertible.• The matrix26641 2 0 03 4 0 05 6 7 08 9 10 113775has positive determinant.• An n × n matrix can have at most n real eigenvalues.• If A is a square matrix such that Am= 0 for some m then 0 is an eigenvalue of A.• If A is a square matrix such that Am= 0 for some m then 0 is the only eigenvalue of
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