Inverse of a Matrix
The inverse of a square matrix A is another matrix denoted A^-1 such that A × A^-1 = I, where I is the identity matrix. Only square matrices with a non-zero determinant are invertible. Finding the inverse of a matrix is vital in various applications, including systems of equations, computer graphics, and in understanding linear transformations.
This calculator computes the inverse using Gauss-Jordan elimination. Input the matrix as rows separated by a semicolon (;) and values separated by commas (,). For example, the 2×2 matrix can be represented as follows:
1,2;3,4Understanding how to find the inverse of a matrix will enhance your problem-solving skills in various mathematical fields. The ability to calculate matrix inverses will also help in fields like physics and engineering.
How to Calculate the Inverse of a Matrix
The calculator parses the input into a numeric matrix, validates that it is square, and performs Gauss-Jordan elimination to compute the inverse. The determinant is also computed as part of the process; if the determinant is zero (or extremely close to zero), the matrix is singular and not invertible. It's essential for students and professionals to ensure matrices are invertible before attempt to compute inverses.
For a 2×2 matrix [a b; c d], the inverse can be computed directly as:
det(A) = ad - bc
A^-1 = (1 / det(A)) * [d -b; -c a]
Examples of Matrix Inversion
Example 1
Input: 1,2;3,4
Output (Inverse): -2, 1; 1.5, -0.5
Example 2 (3×3)
Input: 2,0,1;1,3,2;0,1,1
Try this in the calculator to see the result and entry magnitudes chart.
Example 3 (4×4)
Input: 1,0,2,1;0,1,1,0;2,1,0,1;1,0,1,2
This example illustrates the complexity that can arise with larger matrices.
Frequently Asked Questions About Inverse Matrices
What format should I use for input?
Use semicolons to separate rows and commas to separate values within a row, e.g. 1,2;3,4.
Why does it say the matrix is singular?
A matrix is singular when its determinant is zero (or extremely close to zero). Singular matrices do not have inverses, which means they cannot be used in calculations requiring matrix inversion.
Is there a size limit for matrices?
This calculator performs numerical elimination in the browser. While small to medium matrices (2×2, 3×3, 4×4, etc.) work fine, very large matrices may be slow or numerically unstable.
How is this calculator different from others?
Our Inverse Matrix Calculator is designed to provide clear and accurate results using the Gauss-Jordan elimination method, making it easy to visualize and understand the inverse operations.