3. Let A be a lower triangular matrix with elements a;j. Then a;j = 0 when O izj O isj O i>j Oi

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

A: Sol

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: 5: The Hessian matrix for f(x) = x³x3 + x2x3 at the point [1 2 3] is given by: [6 [18 0 31 0 671 20…

A: Let's solve.

Q: Given the bases B={u , Uz} and B' = {u'1, u'z} for R, where U2 and u' u'1= u'2 !! The transition…

A:

Q: Let A and B be two matrices of size 4 x 4 such that det(A) 1. If B is a singular matrix then…

A:

Q: 2. Show that if A is an n x n real symmetric matrix then (Ax) y = x (Ay) for all x, y E R".

A:

Q: Given the bases B={u1 Uz} and B' = {u'ı, u'2} for R?, where 2 and u'1= 1 u' 2 U2 The transition…

A:

Q: Given the bases B={u , u2} and B' ={u'1, u'2} for R, where 1 u' 2 and u '1 , U2 %3D -1 from B to B'…

A:

Q: Given the bases B={U , u2} and B'={u'1, u'2} for R2, where and u'1= u'2 = The transition matrix Pa-g…

A: The given bases are B={u1,u2} and B={u1',u2'} where u1=10  ,u1=01   and  u1'=-10  ,u1'=01 . To find…

Q: Let A 1B = 5 l, and C-1 = If X is a matrix that satisfies: %3D %3D 2B = Al2 - XC then X is

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: 6. Let B be n x n matrix. Define a function T : Mn(R) → M„(IR) by T(X) : XB – BX for all X € M, (R).…

A: A, B and C be n×n matrices, then  A(B + C) = AB + AC (B + C)A = BA + CA

Q: r every square matrix A = (a;;) € Μπ(C) η |4||, = sup ||Ax|t Σ = max Elai TEC ||z||1=1 π Alloo sup…

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: Find the Jacobi matrix for the following function. 5x + 6у f(x, y) = 5x - 4y? Jacobi matrix, (Df)(x,…

A: Given : f1(x, y) = 5x+6y f2(x, y) = 5x²-4y² To find : Determinant of jacobian matrix

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: b) Prove: Let A be an m × n matrix, R" = NS(A) RS(A).

A: Given: Let A be an m×n matrix , Rn=NSA⊕RSA

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: Let A be an m × n matrix with full column rank; that is with rank(A) = n. Find (the unique) r € R"…

A: This is a problem of a linear transformation, vector space.

Q: Solve for the values of x and y using LU Decomposition from the given matrix and prove that PA-LU or…

A:

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: Given the bases B={u , u2} and B' ={u'1, u'z} for R, where U2 and u' 2 The transition matrix PR-B…

A:

Q: 3. Let A be a lower triangular matrix with elements a;j. Then a;; O when %3D O izi O isi O i>j O i<j

A: Let A be a Lower- Triangular Matrix. Then, A=100210431, A is any lower -triangular matrix.

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 an m x n matrix. Show that: ( a ) Ifxε Ν (Α'Α), then AxE R(Α) Ω Ν (A"). (b) N (AT A) =…

A: Let A be an  m×n matrix. Show that: (a) If x∈NATA, then Ax∈RA∩NAT (b) NATA=NA. (c) A and AT have…

Q: Let B be an n X n matr ix and let E be an n X n elementary matrix. Then det(EB) = (det E)(det B)

A:

Q: نقطة واحدة Show element equal to 18 in the * matrix Y Y= 2 5 7 4 8 3 10 20 30 12 14 16 18 Y(18) O…

A:

Q: If A is an m × n matrix of rank r, what are the dimensions of N(A) and N(AT )? Explain.

A:

Q: 4. Given B1 and B2 as the bases of R?. B, = {[).} {HE) and B2 = · Find a. The coordinates of…

A: If a vector is written as, a= c1v1+c2v2, then c1,c2 are the coordinate of that vector a. Then method…

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: Given the bases B={u , U2} and B' ={u'1, u'2} for R2, where and u'1= u' 2 U2 from B to B' is: The…

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: LetY∼Nn(0,Σ) and let A be any nxn symmetric matrix of rank r. Prove that Y′AY∼ χ2 if and only if…

A:

Q: Let A be an m×n matrix, X an n×r matrix, and B an m × r matrix. Show that AX = B if and only if Axj…

A:

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: If the extended st ress at the point p in the level and at i+ j+ k is given by matrix 10 2 222 the…

A: Solution: Matrix is 102222231 and n^=13i+13j+13k

Q: Show that if A is an n x n symmetric matrix, then |(Ax)•y = x•(Ay) for all x, y in R".

A: In symmetric matrix, transpose of the matrix is equal to the matrix. AT=A

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

A: Sol

Q: Let A be an m x n matrix. Is the set K defined below convex ? Why or why not ? K = {y € R" : 3r E R"…

A:

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

A:

Q: Let O be a F.M. of the system Y'=AY on I, where A is 2×2 matrix. Then, . is also a 2 F.M. a) True b)…

A:

Q: 1. Identify the two-by-two matrix with matrix elements a;j = i – j. O (1 (금) ㅇ(9) (오 ㅇ (? )

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Q: Let A = [a;;] be the n x n matrix defined by aji = k and a;; = 0 if i + j. Show that if B is any n x…

A: Given: A=aijn×n,     aii=k and aij=0 if i≠k. Let B be a square matrix of order 3 which can be…

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 square matrix. I f A"= Onwa for %3D n> 2. Then A= Onun

A: we will see example and conclude.

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

A:

Q: Let A be a 6 x 5 matrix. What must a and b be in order to define T : Rª → R* by T(x) = Ax? II

A:

Q: Let A be the n × n matrix given by Show that det (A) = (a - 1) n − 1(a + n - 1)

A: