6.6. Let A and B be unitarily equivalent n x n matrices. a) Prove that trace(A*A) = trace(B*B). b) Use a) to prove that n n ΣΑ Σ.. j,k=1 j,k=1 c) Use b) to prove that the matrices (i 1) 1 2 4 and 2 i are not unitarily equivalent.

Q: 9. True or False (T/F): If True, Prove it; If False, give a counter-example. a) For all nonzero nxn…

A:

Q: 3| |1 Let A 3 1 4 3 1 3 41 1) Verify that: AadjA = det(A) I (Where I is the identity matrix) 2) Find…

A: given matrix isA=1331431341) to verifyA.adj(A)=det(A).I2) to find A-1

Q: 21. Suppose the last column of AB is entirely zero but B itself has no column of zeros. What can you…

A: Introduction: Matrix multiplication, also known as matrix product and the multiplication of two…

Q: Prove the distributive property of matrix multiplication over vector sums for a given matrix A E…

A:

Q: Prove Theorem 1.9. Uniqueness of an Inverse. Let A be an n x n matrix. a) If C and D are matrices…

A:

Q: Assume A is nonsingular. Prove that det(A-1) 1 det(A) Let c be any scalar. Prove that det(cA) = c"…

A:

Q: 3 – i -2+i 1 a) P = Ye= -2+i b) P = Yes No -i 0

A: Unitary matrice.

Q: Let A be an n ×n matrix. Prove that dim(span({In,A,A2, . . .})) ≤n.

A:

Q: Let E and F be the elementary matrices obtained by doing R¡ → R¡ + kR¡ and Rụ → R¡ + K/K…

A: Let  and  be the elementary matrices obtained by doing  and  respectively, where .We need to show…

Q: 3. a) A = 26 i) Find the QDQT factorization of A

A:

Q: Let u = Let b = 13 - 1 a C and v= 13 1 Show that a C is in Span {u, v} for all a and c. Find the…

A: The given vectors  and .We have to show that the vector  is in the span for all .

Q: Consider the matrices, 1 i V1 – 6² 0 -6 A = 1 а 1 B = 1 1 0 V1– 62 a where a and b are real…

A: Given, the matrices                              A=a1i1a1-i1a and B=1-b20-b010b01-b2 Then, we have…

Q: Consider the following block matrix: N a aT c where N is a m ×m real symmetric matrix, a is a…

A: We will use the basic knowledge of matrices to answer this question.

Q: 10) Let A E Mnxn be given by 0 0 ao ... -1 .. A = -1 0 a2 .. 0 0 0 • -1 an-1 .. Compute XA(t).

A:

Q: Prove that the matrices and a a a 0. a 3. a are Similar ( These 2 matrices are not deagonalısable)

A:

Q: Let A be an n by n matrix. (a) Show that Nul(A - I) C Col(A). (b) If A² = A holds. Prove that Col(A)…

A: To show that Nul(A - I) is a subset of Col(A), we need to prove that every vector in Nul(A - I) is…

Q: Suppose u1, ... , Un and v1, ... , Vn are orthonormal bases for Rn. Construct the matrixA=UI:VT that…

A:

Q: be the set of all vectors of the form 4r – 6s – 6t | r + 3s – 15t - 11r + 7s + 8t 14r + 11s 6t with…

A: We have to solve given problem:

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.