3. Determine whether A is diagonalizable and, if so, find an invertible matrix P and a diagonal matrix D and verify that AP=PD. [1 0 1] a. A = 0 1 1 li 1 o]

Q: An nx n matrix A is invertible if and only if it is row equivalent to I. O True O False

A:

Q: Suppose A is n x n invertible matrix ( AA-1 = I), then the first column of A-1 is orthogonal to the…

A:

Q: Let A be an m × n matrix. Show that if x ∈ N(ATA), then Ax is in both R(A) and N(AT ).

A:

Q: Given a 3 x 3 matrix in terms of a scalar c and that det(A) = 8 , find the value(s) of c if 2 -1 3 2…

A:

Q: Given the partitioned matrix A = [ B]. B]₁ C D in which B and C are invertible. Which of the…

A:

Q: In the given question find an elementary matrix E such that B=EA.

A: given,       A=21-13  ,  B=-1321

Q: .

A:

Q: If A is an m · n matrix where A· A is defined, and k e N, recursively define A*

A: If A is a m × n matrix where A • A is defined and k lies in N then we have to recursively define Ak…

Q: Suppose that A is a singular matrix and A is A = [ k 2 0 0 (k+1) 2 0 1 k ] Find…

A: Singular Matrix

Q: 5. Compute the condition number K(A) of the following matrix 1 1+€ 1 1 1 A 1 1 1 1+€ What is k(A) as…

A: Conditional number is κ∞(A)=∥A∥∞·∥A-1∥∞.If matrix A is known to us so ∥A∥∞ is easy to find .First…

Q: (2) Consider the matrix -[: ] A = Find the unique x, y E R, such that A? = xA – yI (where I…

A:

Q: 4. Prove that if A is a square matrix, then det(A™ A) = det(AA") %3D

A:

Q: Let A be an m × n matrix. Under what circumstances will ||A||2 = ||A||F?

A:

Q: Suppose A is a 2 x 2 matrix such that A and 1 A What is A-1?

A:

Q: Let A be a m times nm×n matrix such that Ax​=b and Ax​=c each have a solution for some choice of b…

A: Let A be an m×n matrix such that Ax​=b and Ax​=c each have a solution for some choice of b and c in…

Q: 1. Find a condition on a, b, c, d such that the augmented matrix " E b has a unique solution.

A: The condition on a,b,c,d must be a relation between a,b,c,d.

Q: 3. Suppose A E Mnxn(C) is a normal, invertible matrix. Prove that A* A-1 is unitary.

A:

Q: Suppose A is an n×n matrix ,and let v1,...,vn ∈Rn. Suppose{Av1,...,Avn} is linearly independent.…

A: It is given that {Av1,...,Avn} is linearly independent. We will prove that A is nonsingular.

Q: Show that if A is any m × n matrix, then AAT and AT A are symmetric.

A: There is a matrix  A given of order m×n. To show that AAT  and  ATA  are symmetric. Solution: The…

Q: Prove If A is an invertible matrix, then AT is invertible and (AT)-1=(A-1)T

A:

Q: compute e^At and e^-At of an identical matrix,I (1  0  0  1)

A:

Q: Let A be an m × n matrix. Prove that if AT A is nonsingu- lar, then rank A = n.

A: Given that, A is an n x n matrix. A^T is the transpose matrix of A. AT A is non-singular. We have to…

Q: Let A be an nxn matrix. If A-= det A adj A is computed, what should AA be equal to in order to…

A: Using the given matrix equation we need to find the value of AA-1 .

Q: Let A ∈ Mn × n (R) such that A3 = In × n and A≠In × n Give an example of a 3 × 3 matrix that…

A:

Q: Suppose A is a matrix that is 0 0 0 0 0 0 0 0 0 What is I-A

A:

Q: Find a 3 × 2 matrix T, such that TT' = A, where 4 A = |4 3 3 5

A: Given: To find a 3 x 2 matrix T, such that TT' = A,  A = 540453035

Q: (b) triangular with l's on its diagonal, and U is upper triangular. Use the results in (a) to obtain…

A:

Q: 4. If A is a nonsingular nxn matrix, show that AA¹ is nonsingular and det(AA) > 0.

A: Given A is a nonsingular matrix.

Q: Let A be an n × n matrix with a column of zeros. Show directly from the Leibniz formula that…

A: Let, A be an n×n matrix with a column of zeros.  To prove detA=0 using Leibnitz formula for…

Q: Let A E Mn(C) denote the n x n matrix with all entries equal to 1. Show that A is positive…

A: Let A belongs to  Mn (C) denote the n x n matrix with all entries equal to 1. Show that A is…

Q: In all parts, suppose that A is a matrix of size 11 x 8 (i) Assuming that rank(Atr) = 3, determine…

A: It is given that

Q: 6. Prove that the matrix A = -1 2 is diagonalizable, and find the corresponding diagonal matrix D…

A:

Q: Suppose that A is an n × n matrix and det A + 0. Determine the rank and nullity of A.

A:

Q: 1 Find the real constants a and b in the matrix Q - a such that Q is orthogonal b

A:

Q: 3. If A is a nonsingular matrix, then A' is nonsingular and its inverse is (A')1 = (A') '. Prove it!

A: A matrix A is said to be invertible if and only if  det(A) is not equal to zero i.e.  |A| ≠ 0

Q: Show that for any n x n matrix A there exists another n x n matrix M such that MAM-1 = AT, where AT…

A: Consider the given information. Let the matrix A is an n×n matrix and M is a m×m matrix. And the…

Q: Suppose A is a 2 x 2 matrix such that for all x, y E R², (Ax) · (Ay) = x ·y Must it be true that A…

A: Please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: 1. Find a matrix P, that orthogonally diagonalizes A,and determine P-1 AP 3. A = %3D 1

A: First we calculate eigenvalues and eigenvectors of matrix A

Q: Let A = 3 0 -1 2 Compute the indicated matrix. (If this is not possible, enter DNE in any single…

A: Solve  for the A^3

Q: 4. Find a diagonal matrix A that satisfies the given condition. 16 At = 0 81 0 1. 01

A:

Q: Show that if A is an n × n matrix, then det(AAT ) ≥ 0.

A: Given, A is a n x n matrix. Let AT be the transpose of A. To prove det (AAT) is non-negative.

Q: Suppose that A e Mnxn(R) is a positive-definite symmetric matrix. Since A is symmetric, A is or-…

A: We will use the knowledge of standard value decomposition.

Q: Let A be a nilpotent matrix (that is, Am = 0 for some m > 1). Prove that if A is diagonalizable,…

A: Consider a nilpotent matrix A. So Am=0 for some m>1.

Q: O Let A be an m x n matrix, and let B be the result of performing a single elementary row operation…

A:

Q: Let 6 A = E = F = 4 2 Compute the indicated matrix. (If this is not possible, enter DNE in any…

A:

Q: Consider 1 A = 2 2 1 1 3 -1 -3 4 Convert A to an upper triangular matrix U by using Gaussian…

A: We have to convert the matrix A=121213-1-34 into an upper triangular matrix U by using Gaussian…

Q: Decide whether the following statement is true or false. Here A is an n × n matrix. If the equation…

A: To find whether the given statement is true or false

Q: 4. (a) Prove that if A is an n x n invertible matrix that satisfies the equation A2 = A then A is…

A: Given [A] is n×n invertible matrix.  And satisfies A2 = A a) To prove [A] is an identity matrix b)…

Q: Suppose that A is a 4 x 4 matrix such that det A = 10. Enter the value of det(A³).

A:

Q: 21 1 2 1 ol -1 A = | 0 2.

A: