Suppose n, k ≥3 are integers with n ≤2k/2. Prove there exists an n ×n matrix A filled with 0’s and 1’s such that no k ×k submatrix is all 0’s or all 1’s. (a k ×k submatrix is a matrix formed by choosing any k rows and any k columns of A). Hint: Use the probabilistic method.

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

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Q: 4. For whichn= 4k, there does not exist a Hadamard matrix of order n.

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Q: Show that an n x n nonnegative matrix is transitive if and only if £=1 4* is a positive matrix. -k=1

A: Given that, An nonnegative matrix is transitive.…

Q: (c) Determine the standard matrices [â] och [T] for â& and Tin the base Sn (for the appropriate n).…

A: Given: A mapping T:ℙ2→ℙ2 defined by, T(p)=p'+p and the mapping x^:ℙ2→ℙ3 defined by, x^(p)=xp. To…

Q: Let A and B be symmetric n × n matrices. For eachof the following, determine whether the given…

A: Let A and B be symmetric n × n matrices. Then We have to prove that F=ABA is also symmetric, for…

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

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Q: 10. Suppose that A is an n x n matrix with real entries such that the diagonal elements are all…

A: The problems with proof can be solved by using the concepts in linear algebra. A diagonally dominant…

Q: 8. Let (:) a T = c d be a 2x2 matrix with real valued entries. Prove that there exists M E R such…

A: To Prove- Let T = abcd be a 2×2 matrix with real valued entries. Prove that there exists M ∈ ℝ such…

Q: Suppose A is a 2×2 matrix and Bis a 3×2 matrix. Select all of the following expressions that are…

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Q: 8. Give an example of a nonzero matrix A such that that Aª = 0 for a positive integer n. Show that…

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Q: Let A and B be n × m and m × n matrices respectively. If AB = In and BA = Im prove let m = n. Where…

A: Invertible  matrix problem....

Q: An n*n matrix C is said to be symmetric if C Select one: True O False

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Q: Let Q be the n × n real matrix in which each entry is 1. Show that det(Q−nIn ) = 0.

A: Given:         Q=11....111....1....................11....1n×n,…

Q: A Hessenberg matrix Hn is n x n matrix that has 2 on the main diaonal, 1 everywhere below the main…

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Q: 8. Let T-(::) a c d be a 2x2 matrix with real valued entries. Prove that there exists M ER such that…

A: We use some useful inequalities.

Q: Let A and B be symmetric n × n matrices. For eachof the following, determine whether the given…

A: Given that A and B is a symmetric matrix: Here the objective is to determine whether D is symmetric…

Q: Find a matrix A such that [is: is a solution for AX=X'.

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Q: Fond the imnerse ķ a) A = 3 6) As Vsing Gamss- Jordan Mrethats

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Q: Suppose that A is an m×n matrix in which no two columns are identical. Prove that A∗A is a diagonal…

A: Given that A is an m*n matrix in which no two column are identical.Since A*A is defined only if A…

Q: Let A be a m × n matrix with columns c1, …, cn. Prove that {c1, …, cn} is independent if and only if…

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Q: If Q+A= A holds for every m×n matrix A, show that Q = 0mn.

A: Consider the given equation, Q=A+A   Now, subtract the matrix A from both side: Q+A-A = A -A As, A…

Q: 5. For a random vector X = (X1, ..., Xn), we define its covariance matrix to be the n × n matrix C…

A: Given that  X=X1, ..., Xn be a random vector with covariance matrix Cij=EXi-X¯iXj-X¯j where X¯i=EXi…

Q: Show that if An = O for some positive integer n (i.e., if A is a nilpotent matrix), then det(A) = 0.

A: We have An=0. So xn is an annihilating polynomial of the matrix. So the minimal polynomial is xm for…

Q: 13. Find an estimate for det (A) if A is an n × n matrix whose entries form some pattern of +10's

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Q: Let A be an n × n matrix. Prove that if Ax = x for all n × 1 matrices x, then A = In.

A: If we have :                     A as an n×n matrix;   To prove:                            If Ax=x…

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: 4. Let M (for matrices) be the set of all 2 x 2 matrices over the reals and x, y ∈AM. Does…

A: This statement is not true.

Q: Let A be a 4 x 4 matrix. Present systematically all cases for the Jordan Canonical Form of A.

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Q: Find K such that the follawing matrix M IS singular. M= | 3. -2 -2 -4 4+K 4 10

A: A matrix is singular if and only if its determinant is zero.   k=?

Q: For every matrix A, it is true that (AT)T = A O True O False

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Q: Find matrix K such that MKN = O if Show complete solution. 1 M = -2 3 1 2 0 0 1 N= 0 8 6 6-1 -4 0 -2…

A: Given- M=14-231-2N=20001-1O=86-66-11-400 To find- K such that MKN=0

Q: Let M and N be 3 x 3 matrices with det(M) = 64 and det (N) = 81. Calculate det Also show whether NT…

A: Let M and N be are 3×3 Matrices. We have given that ,  detM = 64  and  detN = 81 We need to find ,…

Q: Suppose A is a 1 x 7 matrix and is a 7 x 1 matrix. Select all that apply A+Bexists and is al x 7…

A: Given that A is a 1×7 matrix and B is a 7×1 matrix. Here, we will check which of the given options…

Q: Let In denote the n×n identity matrix and On denote the n×n zero matrix.  True or False: imIn =…

A: We have to solve

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

A: Given: Matrix A of order m × n.

Q: Consider M as an n x n matrix, we can say that M is nilpotent  if Mx = 0 for some x ∈ +Z . Now:…

A: (a) Let M be a nilpotent matrix .  So, using definition of nilpotent, Mx=0 for some x∈ℤ+ ⇒Mx=0⇒Mx=0…

Q: Suppose that the n×n matrices A and B​ commute; that​ is, that AB=BA. Prove that eA+B=eAeB.

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

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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: Suppose that a matrix M is nilpotent if exists an x element of Z+ in where Mx = 0. If M is…

A: We will prove the given result.

Q: For every real nx n matriz A, there is a s either a Householder matrix or the identity, R=H₂H₂H₁A.

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Q: Suppose that A is a 3 x 12 matrix, and that 7(x) = Ax. If T is onto, then what is the dimension of…

A: We know that the null space of a matrix A of order n×m is the number of linearly independent…

Q: Consider the following four 2x2 matrices: Aj = A2 = A3 = and A = Prove that a 2x2 matrix M commutes…

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Q: A matrix M ∈Mn×n(C) is called skew-symmetric if Mt= −M. Prove that if M is skew-symmetric and n is…

A: The matrix is not invertible if its determinant is 0.If k is multiplied to the matrix Anxn then,

Q: Suppose you have an mxn matrix A and its Singular Value Decomposition (SVD): A = UEVT Pick all of…

A: Solution

Q: 2.3 Compute the principal square root of each of the lowing positive semidefinite matrices. (b) 2 3…

A: We shall solve first question only as you have asked more than one Que of same type. For others…

Q: The number obtained by a cofactor expansion of a matrix A is not dependent of the row or column…

A: Given the matrix A  The number obtained by a cofactor expansion of a matrix A it's not dependent of…