Let A be a non-singular matrix of the order 2 × 2 then |A-1|=

The correct option is 1 / |A|.

Let us understand why this is the correct answer step-by-step.

By definition, a square matrix A is said to be non-singular if its determinant is non-zero, that is:
$$\left|A\right|\ne 0$$

For any non-singular matrix A, there exists an inverse matrix denoted by A^-1 such that:
$$A·A^{-1}=I$$
where I is the identity matrix of the same order (2 × 2 in this case).

Taking the determinant on both sides of the equation, we get:
$$|A·A^{-1}|=\left|I\right|$$

We know two important properties of determinants:
1. The determinant of the product of two matrices is the product of their individual determinants, i.e., $|A·B|=\left|A\right|·\left|B\right|$.
2. The determinant of the identity matrix I is always 1, i.e., $\left|I\right|=1$.

Applying these properties, our equation becomes:
$$\left|A\right|·\left|A^{-1}\right|=1$$

Since A is non-singular, we know that $\left|A\right|\ne 0$. Therefore, we can divide both sides by $\left|A\right|$ to solve for $\left|A^{-1}\right|$:
$$\left|A^{-1}\right|=\frac{1}{\left|A\right|}$$

This confirms that the determinant of the inverse matrix is the reciprocal of the determinant of the matrix itself, which matches the option 1 / |A|.