alleryA square matrix $A$ has an inverse $A^{-1}$ if and only if:
A$A = A^T$
B$A$ is symmetric
C$\det(A) \neq 0$
D$A$ has more rows than columns
Answer & Solution
Correct answer: C. $\det(A) \neq 0$
A square matrix is invertible if and only if its determinant is non-zero. Such matrices are called **non-singular**. A matrix with $\det = 0$ is **singular** and has no inverse.
Why: the inverse formula has $\det(A)$ in the denominator: $A^{-1} = \dfrac{1}{\det A} \cdot \text{adj}(A)$, where adj is the adjugate. If $\det(A) = 0$ you'd be dividing by zero.
Symmetry and other properties (options A, D) don't determine invertibility. A symmetric matrix can be singular (e.g. the zero matrix is symmetric but singular).
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