Inverse of a Matrix

Clear explanation of the inverse of a matrix with definitions, conditions, methods, and examples using the Gauss–Jordan method and the adjoint formula.

1. What Is the Inverse of a Matrix?

The inverse of a matrix is a matrix that ‘undoes’ the effect of the original matrix. For a square matrix \( A \), its inverse is a matrix \( A^{-1} \) such that:

\( AA^{-1} = A^{-1}A = I \)

This is very similar to how the reciprocal of a number works. Example: The inverse of 5 is \( \frac{1}{5} \) because:

\( 5 \cdot \frac{1}{5} = 1 \)

Likewise, the inverse matrix gives the identity matrix when multiplied by the original.

2. When Does the Inverse of a Matrix Exist?

Not every matrix has an inverse. A matrix is invertible (or non-singular) if and only if:

  • It is a square matrix.
  • Its determinant is non-zero.

If the determinant is zero, the matrix is singular and has no inverse.

3. Methods to Find the Inverse of a Matrix

There are two main methods taught in school-level mathematics:

  • Gauss–Jordan Method (Row Reduction)
  • Adjoint Method

4. Method 1: Gauss–Jordan Method (Row Reduction)

In this method, we convert the matrix into the identity matrix by applying elementary row transformations. At the same time, we apply the same operations on the identity matrix to obtain the inverse.

Steps:

  1. Write the augmented matrix \( [A | I] \).
  2. Apply row operations to transform \( A \) into \( I \).
  3. The matrix that appears on the right becomes \( A^{-1} \).

4.1. Example

Find the inverse of:

A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}

Form the augmented matrix:

[A | I] = \left[ \begin{array}{cc|cc} 1 & 2 & 1 & 0 \\ 3 & 4 & 0 & 1 \end{array} \right]

Apply row operations:

  • \( R_2 \rightarrow R_2 - 3R_1 \)
  • \( R_1 \rightarrow R_1 - 2R_2 \)

Final form:

\left[ \begin{array}{cc|cc} 1 & 0 & -2 & 1 \\ 0 & 1 & 1.5 & -0.5 \end{array} \right]

Thus,

A^{-1} = \begin{bmatrix} -2 & 1 \\ 1.5 & -0.5 \end{bmatrix}

5. Method 2: Using the Adjoint (Adjugate) of a Matrix

For a square matrix \( A \), the inverse is given by:

\( A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A) \)

This formula works when the determinant is non-zero.

5.1. Steps

  1. Find the determinant of \( A \).
  2. Find the cofactor matrix.
  3. Take the transpose of the cofactor matrix to get the adjoint.
  4. Apply the formula.

5.2. Example

Let:

A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}

1. Determinant:

\det(A) = 8 - 3 = 5

2. Cofactor matrix:

\begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix}

3. Adjoint (transpose):

\text{adj}(A) = \begin{bmatrix} 4 & -1 \\ -3 & 2 \end{bmatrix}

4. Apply formula:

A^{-1} = \frac{1}{5} \begin{bmatrix} 4 & -1 \\ -3 & 2 \end{bmatrix}

6. Checking the Inverse

You can verify your result by multiplying:

AA^{-1} \, \text{and} \, A^{-1}A

If both products give the identity matrix, the inverse is correct.

7. Important Notes

  • A matrix with zero determinant cannot be inverted.
  • Only square matrices can have inverses.
  • The inverse is unique (there is only one inverse).
  • A matrix of full rank always has an inverse.

8. Why Do We Use Inverses?

Matrix inverses are used in:

  • solving systems of linear equations (\( AX = B \Rightarrow X = A^{-1}B \))
  • transformations and geometry
  • finding matrix powers
  • computer graphics
  • mathematical modelling

The inverse plays the same role for matrices as reciprocals do for numbers—it helps us reverse operations.