Let A be an n x m matrix. What is the value of rank(A) + nullity(A)? A) т B) n +1 C) m + n D) т — п E)

Q: Let U be an n × n upper triangular matrix with nonzero diagonal entries. (a) Explain why U must be…

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Q: a Verify that det EA = (det E)(det A), where E is the elementary matrix shown and A =

A: E=1k01  A=abcd

Q: Let A be an n × n matrix with triangular factorization LU. Show that det(A) = u11u22 · · · unn

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Q: Exercise 3 Let A be an n x n matrix satisfying A² = 4A. What are the possible values of det(A)?

A: A be an n×n matrix satisfying A2=4A

Q: Is this the correct definition of upper triangular, for a matrix A? Aij * 0 when j 2 i O Yes No

A: Since you have asked multiple questions , we will solve first question for you. If you want any…

Q: Let A be a lower triangular matrix with elements aij. Then aij = 0 when O izj O isj O i> j O i<j

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Q: Let A, B & Mn (C) be Hermitian matrices. Then

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Q: Let A be an m × n matrix. Prove that N(A) ⊂ N(ATA).

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Q: Show that if A is any mxn matrix, then Im= A.

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Q: let Amxn matrix , c scalar If cA=0, then Select one: a. c=0 and A=0 b. c=0 or A=0 C. C0 or A=0

A: WE have to check the meaning of cA=0.

Q: Prove the theorem: AAT is a symmetric matrix for any matric A.

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Q: 32. Construct a nondiagonal 2 x 2 matrix that is diagonalizable but not invertible.

A: .

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: Let A be square matrix. If A" = 0. A= Oava %3D for n> 2. Then

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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: ) Consider the matrix A = [a b] with a, b, c and d real-valued numbers. Under which condition(s) for…

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Q: Assuming that b # 0, find a matrix that orthogonally diagonalizes a= [: :] P

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Q: Show that if A is a symmetric matrix, then Ak , k = 2, 3,... , is symmetric.

A: Given that, A is a symmetric matrix. Then we get, A^T = A; where A^T is the transpose of matrix A We…

Q: Find an LU-factorization of the matrix A such that A = LU. %3D 6 4 -2 1 -2 3 2. -2

A: Explanation of the answer is as follows

Q: Is it true or false? If A is nxn matrix, c is scalar, then cA |=c | A| Birini seçin:

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Q: Consider an n x n matrix. Give 3 equivalent conditions for A to be invertible

A: Let A be an n×n matrix. we have to give 3 equivalent condition for A to be invertible. Conditions…

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Q: Let A, B, and C be three nonzero n × n matrices such that AC = BC. Does it follow that A = B? If so,…

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Q: Let A be a 4x6 matrix. what is the dimension of Col A if the nullity is 2? Explain

A: 4

Q: Explain why the columns of an n x n matrix A span R^n when A is invertible

A: From the given information:

Q: Find a 2 x 2 matrix B + 0 and B + 1, such that AB = BA, where A How many such matri- 2 ces B are…

A: Given data: The matrix is given by "A". A=1221 Consider the matrix "B". B=abcd According to the…

Q: * Let A be and m x n matrix. Describe the matrix E such that EA is A with row swapped.

A: Given matrix A is a m×n matrix. And the elementary matrix E and EA are row swapped. Since elementary…

Q: Prove If U is a unit upper triangular matrix, then U is nonsingular and U−1 is also unit upper…

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Q: (b) Explain why in general, if u = [u uz] is a row of length 2, A = a1,1 a1,2 [@2,1 @2,2] we always…

A: Answer

Q: Г1 3 2 5 1 [3 4 2

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Q: Prove the Frobenius theorem in the case where A is a 2 ×2 matrix.

A: This theorem is not valid for order less than 3

Q: Let A be an n × n matrix in which the entries of each row sum to zero. Find det(A).

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Q: Let A be an n x n Hermitian matrix and X and Y be n x 1 column vectors. Prove that (AX)- Y- X- (AY)…

A: Prove.

Q: Find the LU tactorization of 2 -4' -4 7 10 18 That is, write A = LU where L is a lower triangular…

A: I have used Gauss elimination method to find LU decomposition

Q: Show that if A is an m × n matrix such that AAT is non- singular, then rank A = m.

A: Let AAT can be non singular only if the columns in A are linear independent and they span the column…

Q: Explain why the columns of an n×n matrix A span R^n when A is invertible.

A: Consider, a matrix A of order n, which is a invertible matrix.Since, the set of vectors

Q: Suppose A is rectangular ( m by n) and S is symmetric ( m by m).(a) Transpose AT SA to show its…

A: Given: Matrix A is rectangular ( m by n) and matrix S is symmetric ( m by m).

Q: Let A € RPXm have rank r. Show that there are matrices C and B such that A = св, where C E RPXT and…

A: Please check the solution in next step

Q: 18. Let A be a 3 x 5 matrix with 2 pivots. What is the dimension of Nul(A)?

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Q: Explain transpose AT of the m x n matrix A?

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Q: Let A € RPXm have rank r. Show that there are matrices C and B such that A = CB, where C e RPxr and…

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Q: Let A be an n x n matrix and let D be a n x n diagonal matrix. Show that AD =…

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Q: Determine the matrix C such that (CT – 21) 1 1 2 2 3

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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: Let the entries of a matrix A = \aij be defined as aij 2i2 – i+j+ g(j), where g is a %3D function of…

A: Given Ais a symmetric matrix therefore aij=aji…

Q: Consider a matrix A, and let B= rref(A). %3D a. Is ker(A) necessarily equal to ker(B)? Explain. b.…

A: The matrix B=rrefA is the Reduced Roe Echelon form of the matrix A. The kernel of a matrix A is the…

Q: Let A be a 4x4 singular matrix. What is the smallest possible dimension of Null Space of A? A) 2

A: Using,  Rank- Nullity Thereom

Q: Let A be p x p matrix such that ATA = A². Show that: (a) tr (A - AT)" (A -A") =0. (b) Show that A is…

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Q: Let A be an m × n matrix.(a) Describe the diagonal entries of AT A in terms of the columns of A.(b)…

A: (a) Describe the diagonal entries of AT A in terms of the columns of A.