Inverse Linear Transformation Calculator

Calculator Form

Settings

Matrix Dimension

Decimal Precision

Matrix A

Transformed Vector y

The calculator solves x = A^-1y when A is invertible.

Example Data Table

Item	Values
Matrix A	[[2,1,0],[1,2,1],[0,1,2]]
Transformed Vector y	[4,8,8]
Recovered Vector x	[1,2,3]
Determinant	4
Meaning	The matrix is invertible, so the original vector exists uniquely.

Formula Used

Let T(x) = Ax, where A is a square matrix. If det(A) ≠ 0, then A is invertible and the inverse transformation exists.

y = Ax
x = A^-1y
det(A) ≠ 0  =>  inverse exists
r = Ax - y
||r|| = sqrt(sum of squared residual components)

The calculator first tests invertibility through the determinant. Next, it uses Gauss-Jordan elimination to build A^-1. It then multiplies the inverse matrix by the transformed vector y to recover the original vector x. Finally, it verifies the result by computing A × x and the residual norm.

How to Use This Calculator

Choose the matrix size from 2 × 2 up to 5 × 5.
Enter the square matrix values in Matrix A.
Enter the transformed vector values in y.
Set the decimal precision for displayed output.
Press the calculate button to recover the original vector.
Review the determinant, inverse matrix, verification vector, and steps.
Use the CSV or PDF buttons to save the result.
Use the graph to compare original and transformed coordinates.

Frequently Asked Questions

1. What does this calculator solve?

It recovers the original input vector x from a transformed vector y when the rule is y = Ax and the matrix A has an inverse.

2. When does an inverse linear transformation exist?

An inverse exists when the matrix is square and its determinant is not zero. That means the mapping is one-to-one and onto.

3. Why is the determinant shown?

The determinant is the fastest invertibility test here. A zero determinant means the matrix is singular, so no unique inverse transformation exists.

4. Why does the calculator show rank and nullity?

They describe structural information about the transformation. Full rank supports invertibility, while nullity shows how many input directions collapse to zero.

5. What is the residual norm?

It measures how closely the verified vector A × x matches the entered transformed vector. Smaller values indicate a more accurate numerical recovery.

6. Can I use decimals and negative values?

Yes. The inputs accept integers, decimals, and negative numbers. The calculations are handled numerically and displayed using your selected precision.

7. Why do I see no inverse result?

That usually means the determinant is zero. In that case, the matrix compresses space and cannot be reversed uniquely.

8. What does the graph compare?

It compares recovered original coordinates with transformed coordinates. This helps you inspect how the matrix changes magnitude and direction component by component.