Enter a 3x3 matrix and find its inverse. Review determinant, cofactors, adjugate, and identity checks. Download results for class, research, projects, or audits instantly.
| Matrix A | Determinant | Inverse A-1 |
|---|---|---|
| [1, 2, 3] [0, 1, 4] [5, 6, 0] | 1 | [-24, 18, 5] [20, -15, -4] [-5, 4, 1] |
| [2, 0, 1] [3, 0, 0] [5, 1, 1] | 3 | [0, 1/3, 0] [-1, -1, 1] [1, -2/3, 0] |
| [4, 7, 2] [3, 6, 1] [2, 5, 1] | 3 | [1/3, 1, -5/3] [-1/3, 0, 2/3] [1, -2, 1] |
For a square matrix A, the inverse is found with this rule:
A-1 = adj(A) / det(A)
The determinant must not equal zero. The adjugate is the transpose of the cofactor matrix.
For a 3x3 matrix, the determinant is:
det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Here the matrix is [a b c] [d e f] [g h i].
A 3x3 matrix inverse is a matching matrix that reverses multiplication. When a matrix A has an inverse, multiplying A by that inverse gives the identity matrix. This identity has ones on the main diagonal. It has zeros in every other position. The calculator helps you see that relationship without hiding the working.
The determinant is the gatekeeper. If the determinant is zero, the matrix is singular. A singular matrix has no inverse. If the determinant is close to zero, the result may be unstable. That is why this calculator includes a tolerance field. It helps you judge whether a very small determinant should be treated as unsafe.
The tool builds minors, cofactors, and the adjugate matrix. A minor is made by removing one row and one column. The cofactor adds the correct sign to that minor. The adjugate is the transpose of the cofactor matrix. Finally, each adjugate entry is divided by the determinant. This gives the inverse matrix.
An inverse should pass a simple test. Multiply the original matrix by the inverse. The answer should be the identity matrix. Small decimal differences can appear because computers round numbers. The calculator shows this verification matrix. It also displays the determinant, cofactor matrix, adjugate matrix, and optional fraction style values.
Inverse matrices appear in linear equations, transformations, graphics, statistics, engineering, and economics. They help solve systems where three unknowns depend on three equations. They also support coordinate changes and model fitting. This page is designed for students and professionals who need a clear trail from input to result.
Enter exact values when possible. Avoid unnecessary rounding before calculation. Use fractions only as a display aid. They are based on decimal conversion. For critical work, compare the output with a trusted algebra system or manual solution. This improves review quality and confidence.
After calculation, you can download a CSV file for spreadsheets. You can also create a PDF report from the displayed result. These options make it easier to submit homework, save notes, or share a calculation record. Always review the determinant before trusting an inverse from rounded data.
It is a matrix that reverses multiplication by the original 3x3 matrix. When both are multiplied, the result should be the identity matrix.
A 3x3 matrix has no inverse when its determinant is zero. It may also be unreliable when the determinant is extremely close to zero.
The determinant shows whether the matrix is invertible. A nonzero determinant means the inverse can be calculated using the adjugate method.
The adjugate provides the arranged cofactor values needed for the inverse formula. Dividing it by the determinant gives the inverse matrix.
Small decimals appear because floating point arithmetic rounds values. They usually mean the result is close to the identity matrix.
Yes. The calculator accepts positive numbers, negative numbers, decimals, and zero. It checks the determinant before returning an inverse.
It is a safety limit for very small determinants. If the determinant is below that limit, the matrix is treated as unsafe to invert.
The CSV button saves spreadsheet-ready values. The PDF button creates a simple report from the result section shown on the page.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.