Determine if the equation Ax = b is consistent for all possible b1 and b2

Question

Let A =
-3	3
3	3
and  b =
b1
b2

Determine if the equation Ax = b is consistent for all possible b₁ and b₂ . If the equation is not consistent for all possible b₁ and b₂ , give a description of the set of all b for which the equation is consistent (i.e., the condition which must be satisfied by b₁ and b₂ ).

Thank you so much for any help :)

Answer 1

Given we are looking to solve Ax = b for a given A matrix and a b vector, we can first determine the determinant of the matrix A to see if a solution is possible...

det(A) = -3*3 - (3*3) = -18 ≠ 0

Therefore a solution can be found provided what vector b represents (b₁,b₂)^T

If (b₁,b₂)^T = (0,0)^T, then we would be finding the null space of matrix A. In general we can apply the systems of linear equations to find...

-3x₁ + 3x₂ = b₁

3x₁ + 3x₂ = b₂

Solving we get -->

x₂ = (b₁ + 3x₁)/3

Pushing into the second equation -->

3x₁ + (b₁ + 3x₁) = b₂

6x₁ = b₂ - b₁

x₁ = (b₂ - b₁)/6

Back-substituting into the first equation gives the solution for x₂

x₂ = (b₁ + 3x₁)/3 --> (b₁ + (b₂ - b₁)/2)/3 --> (b₂ + b₁)/6

therefore the solution vector (x₁,x₂)^T = 1/6 * (b₂ - b₁, b₂ + b₁)^T

Given values of (b₁,b₂)^T you can now determine the solution for (x₁,x₂)^T

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