Reference no: EM133348716
Question 1. Use the determinant to find all values α ∈ C such that the matrix
![694_matrix.jpg](https://secure.expertsmind.com/CMSImages/694_matrix.jpg)
is invertible.
Question 2. Let A ∈ Fm×n and let b ∈ Fm×1. Prove that the linear system Ax = 0 has a nonzero solution x ∈ Fn×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.)