Prove that the LU factorization is unique by considering the possibility that LU = LŨ, for some L, L lower triangular matrices, U,U upper triangular matrices, such that L+ L or U + Ŭ. %3!

Q: Find an LU factorization of the matrix A (with Lunit lower triangular). 4 -8 8 -4 A = 8 - 14 11 -5…

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Q: Show that if A is any m × n matrix, then Im A = A and AIn = A.

A: We know that if A is a matrix of order m x n and B is another matrix of order n x p, then their…

Q: Let A be an n × n matrix and let Q and V be n × northogonal matrices. Show that ||AV ||2 = || A ||2

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Q: Suppose A. B and X are 'nxn matrices such that A and B are invertible, and Which one of the…

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Q: (0) Construct two non-zero 3x3 matrices A and B such that for A its rank is greater than its nullity…

A: two non-zero 3×3 matrices A and B such that for A its rank is greater than its nullityand for B…

Q: Find an LU factorization of the matrix A (with Lunit lower triangular). 4 A= 12 2 -0 12 8 4,

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Q: Is it possible for A3 to be an identity matrix without A beinginvertible? Explain.

A: To explain:It is possible for A3 to be an identity matrix without A being invertible.By definition

Q: 5. Calculate the inverse of the following matrix using Gauss-Jordan elimination. A =

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Q: in the factoriza

A: Given,      A=160210021

Q: VýB × VA

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Q: Find the charo cteristica poly no nomi the matrix A = 3. 3. /- P(x) = 1.

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Q: [1 2 2 3 1 0 1 1 4 4

A: Given matrix is  123101144find LU factorization of a given matrix

Q: TRUE/FALSE Assume that all the given matrices are n × n. If A = O, but Ak = O (where O denotes the…

A: Given:   To Check: A is singular or not Concept: A is singular only when det(A) = 0

Q: Suppose A, B are n × n matrices, B is invertible, and AB=0 (where O is the n x n zero matrix). Given…

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Q: Find a basis and dimension for the all 4 x 4 skew symmetric matrices with a11 = a12 a9 over R.

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Q: Suppose that A+ is a 2 × 2 matrix and that A+ = -A. Show 7.2 that iA is awesome.

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Q: Show that if A is a reversible matrix on an F body, the At transpose is also reversible.

A: Given that A is a reversible matrix. Then, AA-1=I, where I is an identity matrix and A-1 is the…

Q: 4. Decide if there is a three by three matrix A such that A. 1 for all æ, y, z in R.

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Q: Consider the following 2x2 diagnal matricies k  0 0  j If k, j are subsets of {1,2,3,4.....9} how…

A: We have 9 choices to fill place of k.  Also we have 9 choice to fill place of j.

Q: Find the LU factorization of A = 10 -15 -21 That is, write A = LU where L is a lower triangular…

A: Given: 55108-15-21

Q: Compute the LU factorization of each of the following matrices.

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Q: Find a 3 x 3 matrix A such that Az = 10 for all i in R³. A =

A: Given Ax=10x  We have to find 3×3 matrix A  Now

Q: Let A and B be two 4 × 4 matrices with rank(A)<4. Then rank(A²B)<4. Select one: O True O False

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Q: Ч. W is the set of all 2x2 upper triangular matrices in Mzx2 (IR).

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Q: Suppose Q = PCP-1 where P is orthogonal, and C is upper triangular. Prove that if Q is symmetric,…

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Q: A nxn matrix is called nilpotent if there exists an integer p such that AP=On. Show that each of the…

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Q: Consider the full SVD of a matrix A = UEVH, it is incorrect to say that (a) Only V is unique, (b) U…

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Q: Generalize the definition of LU factorization to nonsquare matrices by simply requiring U to be a…

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Q: Prove that the product of two n x n anti-diagonal matrices is diagonal.

A: To prove that the product of two n*n anti-diagonal matrices is a diagonal matrix.

Q: Compute the LU factorization of each of the following matrices.

A: A matrix                                               We have to compute the LU factorization of…

Q: Determine all the n × n− matrix A with the range such that A2 = A.

A: Clearly the zero matrix and identity matrix are the trivial examples of our problems. However there…

Q: Compute the LU factorization of each of the fol- lowing matrices: 3 (a) 1 (b) 2 5 -2 1 1 1 -2 1 2 3…

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Q: Suppose the matrices P and Q have the same rows as I but in any order. They are "permutation…

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Q: 2) Let A and B be two skew-symmetric matrices of the same size. Show that -AB BA is skew-symmetric.

A: Definition of a skew-symmetric matrix: a matrix A equal to the negative of its transpose, i.e., A =…

Q: 4. Show whether that the following matrices are positive define, negative definite or neither A= 2.…

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Q: 8.) Find the LU-factorization of the following matrix 1 2 2 2 3. 1.

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Q: 4. Prove: If A and B are both n x n matrices, then det(AB)= detAdetB. (Only use 2x2 matrices)

A: Given A and B  n×n  matrices to prove  det(AB)=det A det B

Q: 2. Every square matrix A can be decomposed as a product of three matrices A = SDR where S and R are…

A: We shall give an counter example for part (2)

Q: Compute the LU factorization of each of the following matrices.

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Q: rices in the factori

A: Given,          A=160210021

Q: 1. P.ove that matrıx multıplication is associative. Show that the product of two orthogo- nal…

A: The second part of the question is proved in the claim.

Q: Let x and y be square matrices of order n a. Show that if x and y are skew symmetric, then x + y is…

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Q: Calculate an A = LU factorization, where L is a lower triangular matrix that has all 1s in the…

A: Lu decomposition

Q: Suppose A is a symmetric n × n matrix and B is any n × m matrix. Show that BTAB, BTB, and BBT are…

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Q: 43 A and B are B,3 matrices where lAl = -3 and IBl =2. Giren this, what is |-2 AD-2|?

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Q: Consider a 2x2 matrices a  b c  d a,b,c,d are subsets of {1,2,3}. If you were to form a matrix at…

A: Each of a,b,c,d has 3 choices. So there are 34=81 possible matrices.A matrix will not have inverse…