Linear Algebra Question

Question

Let X=[ -6 -9 ]

7 0

Give an example of two 2x2 matrices A and B, neither of which is the identity matrix, such that AB=X.

Answer 1

A = A₁₁ A₁₂

A₂₁ A₂₂

B = B₁₁ B₁₂

B₂₁ B₂₂

AB = A₁₁*B₁₁+A₁₂*B₂₁ A₁₁*B₁₂+A₁₂*B₂₂ = X = X₁₁ X₁₂ = -6 -9

A₂₁*B₁₁+A₂₂*B₂₁ A₂₁*B₁₂+A₂₂*B₂₂ X₂₁ X₂₂ 7 0

Now you have 4 equations and 8 unknowns, meaning there are many possible solutions. One strategy is to arbitrarily choose the elements in one matrix (make sure it has a non-zero determinant) and solve for the elements of the second.

Hope that helps,

Mike