Answer:
-6
Explanation:
You seem to want the value of q₃ in the matrix Q⁻¹.
Matrix inverse
The inverse matrix, Q⁻¹, is the transpose of the cofactor matrix, divided by the determinant. That makes q₃ = -q₂₁/det(Q):
q₃ = -(-2/3)/((-1(-1/9) -(-2/3)(-1/3))
q₃ = (2/3)/(1/9 -2/9) = (6/9)/(-1/9)
q₃ = -6
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Additional comment
The transpose of the cofactor matrix of a 2×2 matrix is the original with the diagonal terms swapped, and the off-diagonal terms negated. Basically, the only math involved in computing the inverse is finding the determinant, and dividing by it.