Which statement is true about a 2x2 matrix A regarding invertibility?

• A. A is invertible iff det(A) = 0.
• B. A is invertible iff det(A) ≠ 0.
• C. A is invertible iff trace(A) ≠ 0.
• D. A is invertible iff rank(A) < 2.

Answer: (B)

A square matrix is invertible exactly when its determinant is nonzero. For a 2x2 matrix A = [a b; c d], the determinant is ad − bc. If this value is not zero, A has an inverse (and you can find it as (1/(ad − bc)) [d −b; −c a]); the columns are linearly independent, so the transformation is reversible. If ad − bc = 0, the transformation collapses a dimension, the matrix is singular, and no inverse exists.

So the statement that A is invertible iff det(A) ≠ 0 is the correct criterion. The other ideas don’t determine invertibility: a zero determinant is the sign it’s not invertible, not invertible; the trace can be zero even for invertible matrices (for instance [0 1; −1 0] has trace zero but det 1 and is invertible); and invertible matrices have full rank, not less than 2.