Solve system by applying gauss-jordan elimination

Reference no: EM133348716

Question 1. Use the determinant to find all values α ∈ C such that the matrix

is invertible.

Question 2. Let A ∈ F^m×nand let b ∈ F^m×1. Prove that the linear system Ax = 0 has a nonzero solution x ∈ F^n×1 if and only if the n columns of A are linearly dependent.

Question 3. Consider the three linear systems of equations:

x₁ + x₂ = -7 x₁ + x₂ = 2 x₁ + x₂ = 2
x₂ + 2x₃ = 10 x₂ + 2x₃ = 3 x₂ + 2x₃ = 4
2x₂+ x₃ = 2 2x₂ + x₃ = 3 2x₂ + x₃ = -1

corresponding to the systems Ax = b, Ax = c, and Ax = d respectively. Solve all three systems simultaneously by applying Gauss-Jordan elimination to the 3 × 6 augmented matrix [A | b c d ].

Question 4. Consider the linear system of equations:

x₁ - αx₂ = 3
x₂ - βx₃ = 0
-γx₁ + x₃= -2

Find conditions on α, β, γ ∈ R such that the system has no solutions, a unique solution, and infinitely many solutions. In the latter two cases, find the solution set. (Note. Your solution(s) may be in terms of α, β, and γ.)

Reference no: EM133348716

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