Inverse of a Matrix

For a square matrix A, its inverse, denoted as A1, is another matrix such that when multiplied with A, it results in the identity matrix I: A×A1=A1×A=I

  • Conditions:
    • The matrix A must be a square matrix.
    • If the determinant of A (det(A) is non-zero, then A has an inverse.
    • If det(A)=0, the matrix A is singular and does not have an inverse.

Calculation of Inverse:

The inverse of a square matrix A is calculated using the formula: A1=1det(A)adj(A)

Where:

  1. det(A) is the determinant of matrix A.
  2. adj(A) is the adjoint (or adjugate) of matrix A.
  3. For 2×2 Matrix: For a matrix A=[abcd], the inverse A1 is given by: A1=1det(A)[dbca] where det(A)=adbc must not be zero.

    Example:

    Consider a matrix A:

    A=[2354]
    1. Determinant: det(A)=(2×4)(3×5)=815=7
    2. Adjoint: From the previous example, the adjoint of A is:
    adj(A)=[4352]
    1. Inverse Calculation:
    A1=1det(A)adj(A)=17[4352]=[47375727]

     

  4. For 3×3 or Larger Matrices: The inverse of larger matrices can be calculated using various methods such as Gauss-Jordan elimination, cofactor expansion, or using software tools.

Properties of the Inverse:

  1. Inverse of Inverse: (A1)1=A

  2. Product with Inverse: A×A1=A1×A=I

  3. Transpose of Inverse: (A1)T=(AT)1

Applications:

  • Solving Systems of Equations: The inverse of a matrix is used to solve systems of linear equations in matrix form (AX=BX=A1B).

  • Transformations: Inverse matrices are crucial in transforming vectors and shapes in geometry and computer graphics.