1. Use the method of elimination to find all solutions to the linear system = 15...

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1. Use the method of elimination to find all solutions to the linear system = 15 5x1 + 3x₂ 3x1 + 5x2 + 3x3 = 0 = 0 3x3 + 5x4 + 3x5 = 0 3x2 + 5x3 + 3x₁ 3x4 + 5x5 =-15 2. Let A and B be any nxn symmetric matrix. Show that A+kB¹ is symmetric for any scalar k. 3. Consider three points (21, 1), (2, 32) and (23,93) in a 2D plane where 21, 22 and 3 are distinct. Show that they lie on some curve with equation y = a +bx+cr² for some scalars a, b and c. 4. Find the condition of c₁ in terms of c₂ and c3 such that the following linear system is solvable: 1 -1 -1 (0-0) 4-2 2 -3 1 5. Let I be a 2 x 2 identity matrix. Define = ^-(1-3). A = 2 (a) By induction, prove that A"=31A+ 33"I for all n € Z+. (b) Does A" converge when n→ ∞0? (c) Define Compute M10 M 4-3 0 0 000 0 1 00 0 0 0 0 0 0 4 -3 0 0 0 00100000 0 0 00004 -3 0 0 0 0 0 0 0000 0 0 00 0 1 0 00 0 0 00 4-3 1 0 1. Use the method of elimination to find all solutions to the linear system = 15 5x1 + 3x₂ 3x1 + 5x2 + 3x3 = 0 = 0 3x3 + 5x4 + 3x5 = 0 3x2 + 5x3 + 3x₁ 3x4 + 5x5 =-15 2. Let A and B be any nxn symmetric matrix. Show that A+kB¹ is symmetric for any scalar k. 3. Consider three points (21, 1), (2, 32) and (23,93) in a 2D plane where 21, 22 and 3 are distinct. Show that they lie on some curve with equation y = a +bx+cr² for some scalars a, b and c. 4. Find the condition of c₁ in terms of c₂ and c3 such that the following linear system is solvable: 1 -1 -1 (0-0) 4-2 2 -3 1 5. Let I be a 2 x 2 identity matrix. Define = ^-(1-3). A = 2 (a) By induction, prove that A"=31A+ 33"I for all n € Z+. (b) Does A" converge when n→ ∞0? (c) Define Compute M10 M 4-3 0 0 000 0 1 00 0 0 0 0 0 0 4 -3 0 0 0 00100000 0 0 00004 -3 0 0 0 0 0 0 0000 0 0 00 0 1 0 00 0 0 00 4-3 1 0