Q4. Prove that the LU factorization is unique by considering the possibility that LU = LŨ, for some L, L lower triangular matrices, U,Ũ upper triangular matrices, such that L+L or U # U. (Hint: you will need HW 2 Q3, and discussion week 2 Q3.) Note: For the LU decomposition, we always choose L such that it has only 1s on its diagonal. You should assume this property for L and L.

Q: Find the LU factorization of 8. That is, write A = LU where L is a lower triangular matrix with ones…

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Q: I The product of two lower triangular matrices is again lower triangular (all its entries above the…

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A: We will first take an arbitrary point (x,y,z) then calculate its reflection point (x',y',z') about…

Q: Prove using elements that for any square matrices A,B of the same size and constant K.…

A: For any square matrices A,B of the same size and constant K.  tr(kA+B)=ktr(A)+tr(B)

Q: b) Evaluate S(2y + 4z)dx + (2æ + 3z)dy + (4x + 3y)dz, where C is the curve given by r(t) = for 0 <t…

A: given ∫C2y+4zdx+2x+3zdy+4x+3ydz where C is the curve given by rt=t3,2sinπt2,3cosπt2 for 0≤t≤1…

Q: eness of the following 3 (a) A = 3 -1 -4 -1 1 -1 6 0 -2 -1 (b) В %— -2

A: RULES: The pivot elements of a symmetric matrix A are the eigenvalues of the matrix A. A is…

Q: 3. Let A = 3-1 1 V₁ = 3 -2 2 = 2 V₂ = 3 V3 = 6 V4 and use the facts Av₁ = V₁ AV₂ = V₂ AV3 = V3 V4 €…

A: 3. Let , and use the facts  to answer the following question.(c) If A is diagonalizable then give…

Q: 5 Let L = and U = Let Aj = -3 A2 = A3 = 2 8 A4 = -2 -2 Which one of the following statements is…

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Q: 10. Consider the following matrices: A = 13 0 1 3-5 1]. B A² BA CA DA 10-1 215 -3 4], C-[|-||---|| 0…

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Q: 8.7.1. Find the singular values of the following matrices: 1 -2 (c) › (_-²-²) -3 6

A: Let be an matrix. Then the singular values of are the square roots of the positive eigenvalues of…

Q: 9. Show that the matrices (6) 65), (8), and (1) generate M2×2(F).

A: We have to show that the matrix  and  generate

Q: Four MBA students take the same six courses at graduate business school. Their final marks in these…

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Q: Which do you find easier? 3 1 (₁ - ( ) -- (3) (a) A = B = -1 1 1₁- (b) A = ()--(²3) B = -2 15 2 2-2

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Q: n If A, B, and C are 2x2 matrices; and det(A) = 5, det(B) = 1, and det(C)=-4 then compute:…

A: Given A, B, C are 2x2 matrix. det(A)=5, det(B)=1 and det(C)=-4. We have to find, det2CTB2C-1ATC-1.

Q: 8. Choose one of the following to complete. Extra credit may be earned if you complete both. a)…

A: Given that A and B is invertible. Then AB-1 exists. Also, AB-1=A-1B-1.

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