Let A be an m x n matrix and be Rm. Suppose u, v E R", with uv, are both solutions to Az = b. Prove that the homogeneous equation Ar = 0 has a nontrivial solution. (Note: You CANNOT assume that b=0.)

Q: 2. Assume that all the operations are properly defined, solve the following equation for the unknown…

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Q: Find a matrix A such that e [-] -2 is a solution for Ax= x'.

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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: Solve For B = 3 the 2 1 20 -3 inverse op matr 1 2

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Q: Let A be square matrix. If A" = 0. A= Oava %3D for n> 2. Then

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Q: Suppose that Q is an orthogonal matrix, then OaQ-8 ObQ-2 d 1Q-0

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Q: suppose (X - XZ) = (Y'Z)". Let X, Y, Z are nxn matrix and X,Z and (X-XZ) are invertible %3D a.…

A: Given, (X-XZ)^-1 =(Y^-1 Z)^-1 X,Z,(X-XZ) are invertible.

Q: 4. Find the symmetric matrix, diagonal form of the quadratic form Q = 2x? + 2y2 + 2z2 – 2xy – 2xz –…

A: Step:-1 Given quadratic form is  Q=2x2 +2y2 +2z2 -2xy -2xz -2yz Q=xyz 2-1-1-12-1-1-12 xyzQ=XTAXHere,…

Q: Inverse Matrix Method.

A: Given equations are x + 2y + z = 2                        ... (1)2x - y - z = 1…

Q: Let B = {b1, b2} and C {C1, C2} be R2. Produce the change-of-coordinate matrix from B to C and from…

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Q: • Let f(u) = a'u = Li=1 a;Ui for a, u E R“. Calculate Vuf(u).

A: Note: We'll answer the first question since the exact one wasn't specified. Please submit a new…

Q: Let J, be the n X n matrix whose entries are all equal to 1, and let D(11, An) be the n X n diagonal…

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Q: Show that if the non-zero vectors s(1), s(2),..., s(m) are conjugate to a positive definite matrix…

A: We are given that the non-zero vectors s(1), s(2), ..., s(n) are conjugate to a positive definite…

Q: Let A =mnd B- then the matrix (2AB + A")" is equal to O None of these O equal to the above 2 17

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Q: 5. Let T: R2 R be defined by [x1 +2x2 T a. Find the matrix [T]B.B relative to the bases B = {u,, u}…

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Q: Find all real numbers a, b for which the matrix A nalizable in R. = ( 1] is diago-

A: We have to find all real numbers  for which the matrix  is diagonalizable in

Q: x 2y * (()) = (2+²) 4x 3. Let T a) T can also be written as T I ((₁)) = 4(3) A b. matrix you found…

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Q: If x=2:2:6, and y=[1, 3, 7]. Write the values of the matrix A=[x' y'] where ' is a matrix transpose.

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Q: 5. Use the matrices to solve the x-2y+3z = 4 system: 3x- 4y+z = 4 2x +3y-4z = -2

A: We have to solve the equation by matrix method.

Q: 1. Write down the 3 × 2 matrix A with aij 2ij – cos 3

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Q: (4) Let A be an invertible matrix. Prove that if A = Q¸R1 decompositions of A and if the diagonal…

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Q: Let A be 3x3 matrix where det(-A) = 2 2007 let B =030 Then det(A BAB) = L004 Answer:

A: Properties of determinant: det(-A)=-det(A)det(A-1)=1det(A)det(AB)=det(A)·det(B)det(An)=(det(A))n If…

Q: Express the quadratic equation in the matrix form x Ax + Kx + f = 0, where a Ax is the associated…

A: The given quadratic expression is .We need to convert this quadratic in the form .

Q: 2. Solve for X in the given matrix equation (A – X) =B where A = 3 2 -1 ol and B = MIN2

A: We have to find X

Q: Find (I – M) and X. 0.7 0.8 40 D = M = 0.3 0.2 65 - 1 Calculate (I – M) '. Select the correct choice…

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Q: If A is 4 by 4 and invertible, describe the nullspace of the 4 by 8 matrix B = [ A A].

A: Given that A is 4×4 and invertible matrix. To describe the null space of the 4×8 matrix B=AA. Null…

Q: Let A be a 3x3 matrix where A2=I3, then det(A) is 1 or -1. Select one: 1-True 2-False

A: To find- Let A be a 3×3 matrix where A2 = I3, then detA is 1 or -1. True False