Given the nxn matrices A,B,C of real numbers, which satisfy the Condition:
A+B+C+λΑΒ=0
Α+Β+C+λBC=0
A+B+C+λCA=0
for some λ≠0 ∈ R
(α) Prove that I+λΑ,Ι+λΒ,Ι+λC are invertible and AB=BC=CA.
(b) Prove that A=B=C
Given the nxn matrices A,B,C of real numbers, which satisfy the Condition: A+B+C+λΑΒ=0 Α+Β+C+λBC=0 A+B+C+λCA=0 for...
Let α and β be real numbers with 0 < α < βく2m and let h : [α, β] → R>o be a continuous function that is always positive. Define Rh,a to be the region of the (x,y)-plane bounded by the following curves specified in polar coordinates: r-h(0), r-2h(0), θ α, and θ:# β. 3. (a) Show that (b) (c) depends only on β-α, not on the function h. Evaluate the above integral in the case where α = π/4...
Determine if the statements are true or false. 1. If A and B are nxn matrices and if A is invertible, then ABA-1 = B. ? A 2. If A and B are real symmetric matrices of size nxn, then (AB)? = BA 3. If A is row equivalent to B, then the systems Ax = 0 and Bx = 0 have the same solution. ? A 4. If, for some matrix A and some vectors x and b we...
Help! 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)
Let α, β, γ ∈ ℝ designate pairwise different real numbers and understand the ℝ-vectorspace P3(ℝ) of real polynomials of degree 2 or less as an inner product space via. = p(α)q(α) + p(β)q(β) + p(γ)q(γ). Now let λ ∈ C / ℝ designate a complex number which is NOT a real number. Question: Show that for every p, q ∈ P3(ℝ) it holds that is a real number. (Hint: show that the number doesn't change through complex conjugation. (NOTE:...
determine whether the given set of invertible n × n matrices with real number entries is a subgroup of GL(n, R).... The set of all n × n invertible symmetric matrices. That is, the set of all matrices where A^T = A and det(A) notequal 0. [Important things to note are that (AB) ^T= (B^T)(A^T) and (A^T ) ^−1 = (A^−1 ) ^T .]
(L33*) Verify the following commutation relations (a) [AB,C] A[B,C +IA,CIB (b) [A,|B C]l-IA,BC] [A,CB] The commutator [..] is an important operation in quantum physics. If the elements A.B.C... satisfy the following conditions, they are said to form a Lie-algebra (ab,c are real or complex numbers): ii) [A,Bl -IB.Al iii) The Jacobi identity [AJB CII + [BJCA]] + [CJA,BII :0 - BA Prove the Jacobi identity for [A,B] AB (L33*) Verify the following commutation relations (a) [AB,C] A[B,C +IA,CIB (b) [A,|B...
PLEASE PROVE PARTS a and b by CONTRADICTION and solve for c as well! Could you explain your steps as well 2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any lER, we can write A = XI + (A - XI) (b) (10 marks) Suppose V is a proper subspace of Mnn(R). That is to say, V is a subspace,...
Numbers 3,4,11 a. SublactiTlnb b. division of nonzero rationals c. function composition of polynomials with real coefficients d. multiplication of 2 × 2 matrices with integer entries e. exponentiation of integers 3. Which of the following binary operations are commutative? a. substraction of integers b. division of nonzero real numbers c. function composition of polynomials with real coefficients d. multiplication of 2 × 2 matrices with real entries e. exponentiation of integers 4. Which of the following sets are closed...
True False a) For nxn A, A and AT can have different eigenvalues. b) The vector v 0 cannot be an eigenvector of A. c) If λ's an eigenvalue of A, then λ2 is an eigenvalue of A2. True False d) If A is invertible, then A is diagonalizable. e) If nxn A is singular, then Null(A) is an eigenspace of A. f) For nxn A, the product of the eigenvalues is the trace of A. True False g) If...