Suppose A is a 5 by 3 matrix and Ax is never zero (except when x = 0). (a) What can you say about the columns of A? (b) Show that AT Ax is also never zero (except when x = 0) by explaining this key step: If AT Ax = 0 then obviously x" AT Ax = 0 and then (WHY?) Ax = 0. (c) Given that AT A is invertible. (i) Explain why B = (AT A)-' AT is a one-sided inverse of A (which side of A?). %3D (ii) Explain why B is NOT a 2-sided inverse of A.

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A: Image is attached with detailed solution.

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Q: please answer part (b)

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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

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Q: For which value of x is the following matrix not invertible: 1 1 3x A = 3x 3x 3x 3x 3x O 2 4 O1/3 1

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Q: Please solve this linear algebra problem

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A: Prove the following (a) Let A and C are symmetric matrices, then . (b)

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A: We need the property (CD)-1 =D-1 C-1  For any two matrices C and D.

Q: Given. Q(x)=3x² + x} +8x? –2x,x, –4.x,x3 Q(x)=x" Ax wher A is symmetric matrix 1 a, = a, = 3…

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Q: 1. Show that the matrix 2 1 A = 1 2 -1 -1 is positive definite.

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Q: = 5.8. Let f₁(x) = x³ x² +1, f₂(x) = x³ + x² + x and f3 (x) = x³ + 2x²+3x+1 be three polynomials in…

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Q: (b) Suppose that A is a square matrix with the property that A³ = 0 (here "O" means the zero matrix…

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Q: (a) A (b) A || = -3 1 2 3 1 0 02 0 0 0 2 (c) A = = || [3 1 0 020 0 1 2

A: Solving for diagonalization of given matrices

Q: (1) Show that I + Y X is also invertible. (2) Show that Y (I + XY )−1 = (I + Y X)−1Y .

A: This is the problem of Linear Algebra, Linear Transformation. Given: X is a m×nmatrix and Yis a n×m…

Q: (c) Consider the quadratic form f(x1, x2, x3) = 3(a₁x₁ + a2x2 + a3x3)² = 4(b₁x₁ + b₂x2 + b3x3)². (i)…

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Q: 5. Answer the following (you need to show your work). (a) Give a 2 x 2 matrix A such that A2 = 0 but…

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Q: O If E is an elementary matrix then so is E". O Let A and B be m x n matrices and D be an n x n…

A: We will prove the given statements.