8. Let T-(::) a c d be a 2x2 matrix with real valued entries. Prove that there exists M ER such that for all x = E R? X2 |Tx| < M \x| (Hint: Try M = 4max {|a| , |b| , |c| , |d]} and use the size bounds.) Bonus: Repeat, with T being an arbitrary m x n matrix.

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: Q5. Show that the function from R"xn to R given by ||4||max = max |a¡j| i,j is not a norm by stating…

A: Given: A function from ·max : ℝn×n→ℝ defined by, Amax = maxi, jaij

Q: if A is (m*n) matrix, prove 'rank(A)=n-dim(N(A))

A:

Q: 6. If A is an nxn matrix whose LU decomposition is PA = LU, , prove that det(A) = (-1)' u₁₁ U 22.²…

A:

Q: Suppose that A is an n x n matrix such that A² = 0. a) Show that if 7 E im(A), then 7 € null(A). (In…

A:

Q: Prove that if A is an m x n matrix of rank r, then the dimension of the solution space of Ax = 0 is…

A:

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: IF A is a nonzero n X n matrix such that A 0, then A is singular. %3D Select one: True False

A:

Q: where En is a k-dimensional vector, A e Ckxk is a constant matrix, and B : Z+ Ckxk is a matrix…

A: Given,      tn+1-t¯=∑i=01αi(tn-i-t¯)+∑i=01βi(zn-i-z¯)…

Q: If A is an q x n matrix with orthonormal columns & W be the column space, then AA"y Projwy for all y…

A: Given that A is a q×n matrix with orthonormal columns. Hence there are q rows and n columns in A.…

Q: If A is an nxn matrix with det(A)=0, then the homogeneous system Ax = 0 has infinitely many…

A:

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: If A is an n×n matrix with the property that Ax = 0 for all x ∈ Rn, show that A = O. Hint: Let x =…

A:

Q: Show that if A exists for an n x n matrix A, then A- is unique.

A: Since, you have posted multiple question but according to guidelines we will solve the first…

Q: How many rows and columns must a matrix A have in order to define a mapping from R* into R³ by the…

A: Find the number of Rows and columns in a matrix A.

Q: 11. Let A be an m x n matrix and let x, y e R". Prove that if Ax = 0 and y = ATb for some b e R",…

A:

Q: .

A:

Q: **32. Suppose A is an m x n matrix and x e R" satisfies (ATA)x = 0. Prove that Ax = 0. (Hint: What…

A: Here we use property of modular and dot product.

Q: If A is an n × n matrix, show that (Ak )T = (AT )k for any positive integer k.

A: Matrix: We will solve it by the method of induction:(Ak)T=(AT)k case 1: k=1case 2: k=ncase 3: k=n+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: If A is an q x n matrix with orthonormal columns & W be the column space, then AA" y = Projwy for…

A: If A is an invertible qxn matrix with orthonormal columns &amp; W be the column space, then the…

Q: (i) Use the properties of trace to argue that tr(AʼA) = tr(AA') for any n × m matrix A. [2 0 (ii)…

A:

Q: Let A be a square invertible matrix. Which of the following is not true? Select one alternative: O A…

A: Since given A is invertible matrix, so det(A) is not equal to zero. It is always true. we also know…

Q: 5. Show that if M is a 3 x 3 magic square with weight w, then we can write M as M = Mo + kJ %3D…

A: Given that ,  A       magic square  To show that : We can write as                     Where,  is a…

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

A:

Q: IF A is a nonzero n X n matrix such that A = 0, then A is singular. Select one: O True False

A:

Q: Suppose that one of the vector functions is a constant multiple of the other on the open interval I…

A:

Q: Let A and B be matrices of size p × m and n × q , respectively. Assume that mn = pq . Define R: Mmn…

A: Matrix A is p×m and B is n×q In a m×n matrix number of element will be m.n R is defined as,…

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: Is the following matrix a Hermitian or Skew-He rmitian, and find the 2 3 – 2i 1+i A X = -3 – 2i 2…

A:

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

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

Q: Let A be a nonsingular k x k matrix with k> 1. Show that. det [adj(A)] = [det(A)]*-1

A:

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: 4. Suppose that xa is an approximate solution to the system of Ax = b. Let A, and Tare nonsingular…

A: Sol

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: Prove that the following is false by giving a counterexample. а ь с a b If matrix de fis invertible,…

A:

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: A matrix Q ϵ Mnxn(R ) is called orthogonal if QQt = I . Prove that if Q is orthogonal, then det(Q) =…

A: Given data: The orthogonal matrix is given by "Q". The given condition is  QQT = I.....(1) Now, take…

Q: Suppose V and W are finite-dimensional and T E L(V,W). Show that with respect to each choice of…

A: Suppose V and W are finite-dimensional and T∈LV,W. To show: to each choice of bases of V and W, the…

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

A: Consider a nilpotent matrix A. So Am=0 for some m&gt;1.

Q: 5. Show that if M is a 3 x 3 magic square with weight w, then we can write M as M = Mo+ kJ where Mo…

A: M is a 3 x 3 magic square matrix with weight w. We take weight, w=15 The matrix of M is  Add any…

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.

Q: Show that if A is any n × n matrix, then (a) A + AT is symmetric. (b) A − AT is skew symmetric.

A: Symmetric matrix: Any matrix A is symmetric matrix if and only if it is equal to its transpose that…

Q: Let A be an m x n matrix. Prove that if B can be obtained from A by anelementary row [column]…

A:

Q: For purposes of this exercise, let us define an "X-matrix" to be a square matrix with an odd number…

A: Given that: X=101010101

Q: A

A:

Q: Find a 3×3 matrix A such that  Ax=4x for all x in R3.

A: To find a 3×3 matrix A such that Ax=4x for all x in R³

Q: For A e Mnxn(C), the function ||A||. = max1<ijśn |aij| is a norm on Mnxn(C) but not a matrix norm.…

A: Matrix Norm: Let Mn×n(ℂ) be the vector space of all matrices of order n with entries in the field ℂ,…