Inverse of a Matrix Using Determinants

Student-friendly explanation of how to find the inverse of a matrix using determinants and adjoint with step-by-step examples.

1. When Does the Inverse of a Matrix Exist?

A square matrix has an inverse only when its determinant is non-zero. If:

|A| \neq 0

then the inverse exists. If:

|A| = 0

the matrix is singular and has no inverse.

2. Formula for the Inverse Using the Adjoint

The inverse of a matrix \( A \) is given by the formula:

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

This formula works only when the determinant is not zero.

3. Steps to Find the Inverse Using Determinants

To find the inverse of a matrix using determinants and adjoint, follow these steps:

  1. Find the determinant of the matrix.
  2. Find the minor of each element.
  3. Convert minors into cofactors.
  4. Create the cofactor matrix.
  5. Take the transpose of the cofactor matrix to get the adjoint.
  6. Apply the formula: \( A^{-1} = \frac{1}{|A|} \text{adj}(A) \).

4. Example 1: Inverse of a 2×2 Matrix

For a 2×2 matrix, the formula becomes especially simple.

Let:

A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

Then:

A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}

This is because the adjoint and determinant formulas are straightforward for 2×2 matrices.

4.1. Numerical Example

Let:

A = \begin{bmatrix} 3 & 4 \\ 2 & 5 \end{bmatrix}

1. Determinant:

|A| = (3)(5) - (4)(2) = 15 - 8 = 7

2. adj(A):

\begin{bmatrix} 5 & -4 \\ -2 & 3 \end{bmatrix}

3. Apply the formula:

A^{-1} = \frac{1}{7} \begin{bmatrix} 5 & -4 \\ -2 & 3 \end{bmatrix}

5. Example 2: Inverse of a 3×3 Matrix

For a 3×3 matrix, the process is longer but follows the same steps.

Let:

A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 2 & 3 & 4 \end{bmatrix}

Find the determinant, cofactors, adjoint, and then apply the inverse formula.

5.1. Outline of Steps

1. Compute the determinant using expansion.

2. Find all minors and convert them into cofactors.

3. Form the cofactor matrix.

4. Take its transpose to get adj(A).

5. Apply the inverse formula:

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

6. Why This Method Works

The adjoint method is based on the identity:

A \cdot \text{adj}(A) = |A| I

Dividing both sides by \( |A| \) gives the inverse formula. This is why the determinant must be non-zero—otherwise division is impossible.