How to Calculate a Matrix Inverse?

To calculate the inverse of a square matrix, follow these steps: find the matrix of minors, the matrix of cofactors, its adjoint (transpose of the cofactor matrix), and finally, divide the adjoint by the determinant of the original matrix.

Here's a detailed breakdown of the process:

1. Check for Invertibility

Before you begin, ensure the matrix is square (same number of rows and columns) and its determinant is not zero. A matrix with a determinant of zero is singular and does not have an inverse.

2. Calculate the Matrix of Minors

For each element in the original matrix, find its minor.
The minor of an element is the determinant of the submatrix formed by deleting the row and column containing that element.
Arrange these minors into a new matrix called the "matrix of minors".

3. Calculate the Matrix of Cofactors

Create the matrix of cofactors from the matrix of minors.
Apply a "checkerboard" pattern of signs (+ and -) to the matrix of minors. Start with a + in the top-left corner.
Multiply each element in the matrix of minors by +1 or -1 according to its position in the checkerboard pattern:
```
+ - + - ...
- + - + ...
+ - + - ...
- + - + ...
...
```
The resulting matrix is the "matrix of cofactors".

4. Find the Adjoint (or Adjugate)

The adjoint (or adjugate) of the matrix is the transpose of the matrix of cofactors.
The transpose is found by swapping the rows and columns of the matrix of cofactors. That is, the element in row i, column j becomes the element in row j, column i.

5. Calculate the Determinant

Compute the determinant of the original matrix. Various methods exist, such as cofactor expansion. For a 2x2 matrix:

| a b |
| c d |

The determinant is (ad - bc).

6. Calculate the Inverse

Divide each element of the adjoint matrix by the determinant you calculated in step 5. This is equivalent to multiplying the adjoint matrix by 1/determinant.
The resulting matrix is the inverse of the original matrix.

Formula:

Inverse(A) = (1 / det(A)) * adj(A)

Where:

A is the original matrix
det(A) is the determinant of A
adj(A) is the adjoint of A

Example (2x2 Matrix):

Let's say we have matrix A:

| 4 7 |
| 2 6 |

Determinant: (4*6) - (7*2) = 24 - 14 = 10
Adjoint:
- Swap the elements on the main diagonal (4 and 6)
- Change the signs of the off-diagonal elements (7 and 2)
Adjoint(A) =

| 6 -7 |
| -2 4 |
Inverse: Multiply the adjoint by 1/determinant (1/10)

Inverse(A) =

| 6/10 -7/10 |
| -2/10 4/10 |

Which simplifies to:

| 0.6 -0.7 |
| -0.2 0.4 |

Important Notes:

Not all square matrices have inverses. Only invertible (non-singular) matrices have inverses. A matrix is invertible if and only if its determinant is non-zero.
The inverse of a matrix, when multiplied by the original matrix, results in the identity matrix.

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