Which of the following is the formula for calculating the inverse of the matrix?

The correct option is (1/|A|) adj A.

To understand why this is the correct formula for finding the inverse of a matrix, let us review the definition of a matrix inverse and its derivation step-by-step.

1. Definition of the Inverse Matrix:
For a square matrix $A$ of order $n\times n$, its inverse matrix is denoted as $A^{-1}$. By definition, multiplying a matrix by its inverse yields the identity matrix $I$:

$$A·A^{-1}=A^{-1}·A=I$$

2. The Relationship Between a Matrix and its Adjugate:
A fundamental theorem in linear algebra states that when a matrix $A$ is multiplied by its adjugate matrix, denoted as $\mathrm{adj}\left(A\right)$, the result is a diagonal matrix where each diagonal entry is equal to the determinant of $A$ (denoted as $\left|A\right|$). Mathematically, this is expressed as:

$$A·\mathrm{adj}\left(A\right)=\left|A\right|·I$$

3. Deriving the Inverse Formula:
Assuming the matrix $A$ is non-singular (meaning its determinant is non-zero, or $\left|A\right|\ne 0$), we can divide both sides of the relation by the scalar $\left|A\right|$:

$$A·\left(\frac{1}{\left|A\right|}·\mathrm{adj}\left(A\right)\right)=I$$

Comparing this equation with the definition of the inverse matrix, $A·A^{-1}=I$, we can directly conclude that:

$$A^{-1}=\frac{1}{\left|A\right|}·\mathrm{adj}\left(A\right)$$

Thus, the correct formula to calculate the inverse of the matrix is indeed $(1/|A\left|\right)\mathrm{adj}A$.