3. (12pts) Answer the following questions: (a) Show that if a square matrix A has decomposition A = LLT with L nonsingular, then A is symmetric positive definite. (b) Find (by hand) the Cholesky factor L in the decomposition A = LLT for the 1 1 1 matrix A1 55 Solve the linear system Ax = b where 1 5 14, b = (3, 11, 20).

Q: Q3. If A = [2 4 41 4 2 4 show that A² -8A-201 = 0 where 1,0 are the unit matrix and 14 4 21 the null…

A: (.) Note: '' As per guidelines we will solve the first question. If you want any specific question…

Q: Given X1 + x2 + x3 = 6 2x1 + x2 + x3 = 7 X1 + 3x2 + x3 = 10 Use Gaussian Elimination to find the…

A: The coefficient matrix for the given system of equations is: A=111211131 The constant matrix is:…

Q: 2 3 A=| 4 7 6 11 For the matrix-matrix 13 calculate the LU decomposition A=LU where 1 U13 U1 U12…

A: We have to solve given problem:

Q: Q2: (a) For given matrix A, find a 3 x 3 non-singular matrix P such that p-1AP is diagonal. A = |-3…

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Q: 2. Prove that: k(4+ mB) - kA+(km) B for any mx n matrices A and B, and scalars k and m. (You may not…

A: 2 We have to prove that kA+mB=kA+kmB Where, A and B are m×n matrices and k and m are the scalars.…

Q: Q4. Find the singular value decomposition of the matrix A this to find the SVD of the matrix (ATA)-2…

A: Given matrix is Then,

Q: Find the RREF of the matrix -4 1 -3 من که من 2 3 -1 4 -2 3 2

A: to find the row reduced echelon form of -4       31      - 1-3       42       - 23        2

Q: 2x1- 3x2+ 2x3= 9 3x1- 2x2+ X3 = 4 X1 - 4x2+ 2x3 = 6

A: Given system of linear equations is ,                        2x1-3x2+2x3= 9…

Q: (-1 3 2 Find the polynomial of a square 3×3 matrix A=| 1 0 -2|given that -3 1 P(x) =14+19x– 3x² –…

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Q: Find the triangular factorization A = LU for the matrix          2  4  -6  A=    1  5  3         1…

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Q: Find a singular value decomposition (SVD) of the matrix: 3 2 2 H = 2 3 -2

A: Given Matrix is  H=32223-2 Then SVD of H is given by: H=UDVT Where, U=Matrix of the orthonormal…

Q: h-2 -4 -2 3 -6+3h -9 -5 8 Decompose the matrix A= -3 -1 6 into L and U matrices by using the method…

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Q: [1 0 [9 0 01 c) A-2 = |0 4 0 Lo o 1l 9. Find a matrix A that satisfies a) (3.4) =0 -1 1 |0 0 -1| 2.

A: ANSWER :

Q: Consider the quadratic form ƒ(x1, x2, x3) = x² + x² + 5x² + 2ax1x2 − 2x1x3 + 4x2x3 = x¹Qx

A:

Q: b) Find the QR decomposition of the matrix [2 -2 17 2 1 0 1 1 0 1 72-0

A: To find- FInd the QR decomposition of the matrix 2-21221011101

Q: ind the Lower and Upper Triangular matrix using Cholesky method

A: Here use the Cholesky method to find the lower and upper triangular matrix of the given matrix.…

Q: 3. a) A = 26 i) Find the QDQT factorization of A

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Q: 4 Apply the Cholesky factorization to A = -2 4 A = UTU. U = lo 10 x1 16 Solve A x2 = -88 by first…

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Q: [1 A 2 1 0 and b : 3 2 3 Jse Gauss-Jordan elimination to compute the inverse of the matrix

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Q: 4. Let A be a matrix with LU decomposition 1 0 0 U11 U12 U13 A = |5 1 U22 U23 7 3 1 U33 Describe the…

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Q: (iv) (v) Use the result in (ii) to show that M³ = 7M – 61. Find M³. (b) Find all the values of λ…

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Q: ) Consider the quadratic form Q(x) = x" Ax, where A = 2 2 2 0 2 %3D Write out Q(x) in terms of z1,…

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Q: 1 4 [-1 Form the Matrix Ak? 3.Let Matrix A = 2 3 and k be any scalar b = -4. -3]

A: We need to find Ak.

Q: solve this plz

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Q: :Use inverse matsix (by Gauss Jordan an Elimination) to solve the syálem 2x + By +3 = - 1 %3D 2x…

A: Consider the given equations 2x+3y+z=-13x+3y+z=12x+4y+z=-2 The given equation can be written as AX=B…

Q: Using the matrices. [1 0 1 R1 = 0 1 1 li o 1 [1 1 1] R2= 1 0 1 li 1 Find the matrix that represents…

A: Use the given matrices.

Q: 1. [ , ] A is a (3 x 4) matrix with the only special solution s = to the equation Ax = 0. Find the…

A: Given that s=2418 is the only special solution to the equation Ax=0 where A is a 3×4 matrix. Note…

Q: If a field K is algebraically closed, then every polynomiel with co efficients in k has a root in K.…

A: A field K is said to be algebraically closed if every polynomial with coefficients in K has a root…

Q: Q3 (b): Find an LU decomposition of the matrix -6 31 -7 2 A =

A: Given matrix is A=3-636-72-170 Find an LU decomposition of the matrix

Q: Apply the Cholesky factorization to A = U = 96 -3 9 0 0 0 0 Solve A y = 16 X = 21 12 I3 = followed…

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Q: Exercise 4: Find a row-reduced matrix which is row-equivalent to -(1 + i) -2 2i A = 0 1 -1

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Q: 6. Consider the 3 x 3 matrix B = 1 2 1 0 2 -1 3 5 7 (a) Perform elementary row operations to convert…

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Q: You deposit $2000 in an account with an annual interest rate of r (in decimal form) compounded…

A: Please refer to the solution in next step.