Inverse of a Matrix
2026-02-28 01:42 Diff
https://brightchamps.com/en-us/math/algebra/inverse-of-matrix

To find the matrix inverse, we use two Inverse matrix methods, such as:

  • Determinant and adjoint Method
  • Elementary Transformation Method
     

Determinant Method: The inverse of a matrix is calculated using the determinant and adjoint. The formula for a square matrix A:


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

Where:

A–1 → inverse of matrix A

det (A)→ determinant of A

adj(A) → adjoint of A

If the determinant of A is zero, the matrix has no inverse.

Elementary Transformation Method: In this method, we find the inverse using a step-by-step process:

Step 1: Express the matrix equation as [A\I]. Here, I is the identity matrix, which is in the same order as the matrix A.

Step 2: To convert matrix A into the identity matrix on the left, use only row operations or only column operations. Apply the same step to the identity matrix on the right side.
The matrix B on the right is the inverse of A once you obtain: \(I = B × A\).

Step 3: Use either row operations or column operations exclusively, not both simultaneously.
Using this method, we can quickly determine the inverse of a 2 × 2 matrix.

2 × 2 matrix Inverse

The inverse of a 2 × 2 matrix can be found using a simple method:

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

Let’s say: Since \( |A| = (ad - bc)\)

\(\text{Adj } A = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \)

  Using the formula:

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

So, 

\(A^{-1} = \frac{1}{(ad - bc)} \times \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \)

Hence, the inverse of the 2 × 2 matrix is determined.

Inverse of a 3 × 3 Matrix

In the case of any 3 × 3 matrix:

\(A = \begin{pmatrix} x & y & z \\ l & m & n \\ a & b & c \end{pmatrix} \)

Use the inverse matrix formula:

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

Determinant of Inverse Matrix

The determinant of an inverse matrix is derived by taking the reciprocal of the original matrix’s determinant:

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


Proof (in simple steps):

Step 1: We know, det(A × B) = det(A) × det(B)\(det(A × B) = det(A) × det(B)\)

Step 2: Also, \(A × A⁻¹ = I\) (by inverse matrix property)

Step 3: So, \(det(A × A⁻¹) = det(I)\)

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

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

Step 4: Thus, the determinant of the inverse is the reciprocal of the original determinant, as shown.