If A is an n×nmatrix whose LU decomposition is PA = LU, prove that det(A)=(-1)' u where r is the number of row interchanges performed during row reduction, and ui are the diagonal elements of U.

Q: Let A= 00 b) Say if A is diagonalisable sur Cv H E M3x3(C) such that H-¹AH is diagonal.

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Q: Compute the new position of the square [A(19,19), B(29,19), C(29,29), D(19,29)] after its rotation…

A: ABCD is a square , we need to rotate it clockwise along with centroid.

Q: Find u and usuch that the null space for the matrix U= 3 v= 0 -2 1 1 -3 -15 1 - 9 1 is span{u, }.

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Q: | 1 2 -1 4 | | 3 3 5 0 | | 2 1 1 3 | | 4 2 2 1 |

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Q: 1. Find an orthogonal change of variables that eliminates the cross product terms in the quadratic…

A: Given Q=x12-4x1x2+x22

Q: Use the inner product = aobo+ aıb1+ azbz+ asb; to find , lIlp|], d(p, q), and then find the angle…

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Q: [61] Diagonalize B = (by finding P and D) or explain why B isn't diagonalizable.

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Q: If A and B are orthogonal, is A+B also orthogonal?  Can you demonstrate Algebraically using A + B =…

A: Solution: Given-   A and B are Orthogonal matricesTo show:  (A+B)  is OrthogonalSince, A and B are…

Q: Find an invertible matrix U such that UA R is the [21 1 -1 -1 0] -12 1 1 A = 3 -1 1-2 3

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Q: Let u = (3, 2, x) and v = (2x, 4, x). (a) Find all possible values of x that make u and v…

A: Given thata)We have to find all possible values of x that make u and v orthogonal.b)We have to find…

Q: State why (u,v) =u,³v,² +u₂³v₂² is not an inner product for u = (u,,u,) and v= (v₁,v₂) in R².

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Q: If A is diagonalizable through D= (P^-1) A P then the diagonal factorization of A = P D (P^-1)…

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Q: Show that LU decomposition is not possible for the matrix A, where 1 2 3 A 2 4 1 0 2

A: The given matrix is,A=123241-102

Q: Prove that the set of mxn upper triangular matrices is a subspace of Mmxn(F).

A: We have to prove that -

Q: Explain why v = ( 1, 0, -1) cannot be a row of A and also in the nullspace.

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Q: Show why the product {[x1, x2], [yı, y2]} = xıy2 does not satisfy the conditions for an inner…

A: See the detailed solution below.

Q: Show how v is orthogonal to any linear combination formed by w1 and w2

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Q: Verify that A = U£VT,

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Q: If an n X n matrix B with det (B) + 0, then for each vector b in R", Bx = b_has several solutions x…

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Q: Q) Let A be diagonalizable so that P-1AP = D. Show that A2 must be diagonalizable.

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Q: Use the inner product =p(-2)q(-2) + p(0)q(0) +p(3)q(3) in P3 to find the orthogonal projection of…

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Q: Use the inner product (А, В) — 2а11bi + az1bz1 + aj2b12 + 2а22b22 to find (A, B) for A = -1' B = 4…

A: Question from linear algebra.

Q: Construct two bijections from (0,1) to (1,1/0).

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Q: Determine if the vector product = a0b0 defines a valid inner product on P2(R) for any…

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Q: Apply Gram-Schmidt to (1, 0, −1, 1), (2, 1, −1, 0) , (2, −1, −1, 3) to obtain an orthonormal set…

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Q: Let S be the parallelogram then determine the area of parallelogram by the vectors [h and b2 h/2 [1…

A: The parallelogram is determined by the vectors  b1→=hh2=10251                              (since…

Q: Find an orthogonal change of variables, x = Py, that transforms the quadratic form q(x) = x² +…

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Q: 2. Find a matrix that projects onto the line / 2x - 5y = 0 and find the projection of 6= (1, -2)…

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Q: Compute: gcd(51, 42) Find a pair of integers x and y such that 51x + 42y = gcd(51, 42) (x, y) = (…

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