Which of the below condition is incorrect for the inverse of a matrix A?

The correct option is "A must be singular matrix".

Let us understand why this condition is incorrect for the inverse of a matrix to exist, by looking at the definition and properties of a matrix inverse.

For any square matrix _A of order _n, its inverse _A^-1 is defined such that:
$$A{A}^{-1}=A^{-1}A=I$$
where _I is the identity matrix of the same order.

The formula to find the inverse of a matrix _A is given by:
$$A^{-1}=\frac{1}{\left|A\right|}\text{adj}\left(A\right)$$
where $\left|A\right|$ (or $\text{det}\left(A\right)$) is the determinant of matrix _A, and $\text{adj}\left(A\right)$ is the adjoint of matrix _A.

From this formula, we can analyze the conditions under which the inverse exists:
1. Square Matrix: The determinant and adjoint are only defined for square matrices. Thus, matrix _A must be a square matrix. (This is a correct condition).
2. Non-zero Determinant: Since the determinant $\left|A\right|$ is in the denominator of the inverse formula, it must not be equal to zero ($\left|A\right|\ne 0$) to prevent division by zero. A matrix with a non-zero determinant is called a non-singular matrix. Therefore, _A must be a non-singular matrix. (This is a correct condition).
3. Adjoint Condition: For the inverse to be defined and non-zero, the adjoint of _A must also not be a zero matrix, i.e., $\text{adj}\left(A\right)\ne 0$. (This is a correct condition).

Conversely, a singular matrix is defined as a matrix whose determinant is equal to zero ($\left|A\right|=0$). If _A is a singular matrix, then its inverse $A^{-1}$ does not exist because division by zero is undefined. Therefore, the statement "A must be singular matrix" is incorrect for the existence of the inverse of a matrix.