3x3 Matrix Calculator – Determinant, Trace, Transpose and Inverse
3x3 Matrix Calculator
Enter the values of your 3x3 matrix below to instantly analyze its determinant, trace, transpose, and inverse.
Quick answer
A matrix calculator performs matrix operations for you. Enter the nine numbers of your 3x3 matrix and click Analyze Matrix. The tool returns the determinant |A|, the trace tr(A), the transpose Aᵀ and, when the determinant is not zero, the inverse A⁻¹.
Our free 3x3 matrix calculator is built for linear algebra homework, engineering checks and computer graphics. It works with whole numbers, negatives and decimals, and Clear Grid resets everything in one click.
A matrix has an inverse only if its determinant is non-zero. If the determinant is 0, the matrix calculator tells you the matrix is singular instead of returning an inverse.
Matrix Formulas Used
- Determinant: |A| = a(ei − fh) − b(di − fg) + c(dh − eg) for the matrix [[a, b, c], [d, e, f], [g, h, i]].
- Trace: tr(A) = a + e + i, the sum of the main diagonal.
- Transpose: rows become columns, so element (i, j) moves to (j, i).
- Inverse: A⁻¹ = adj(A) ÷ |A|, where adj(A) is the transpose of the cofactor matrix.
Worked Example
For A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]:
| Result | Value |
|---|---|
| Determinant | 1(0 − 24) − 2(0 − 20) + 3(0 − 5) = 1 |
| Trace | 1 + 1 + 0 = 2 |
| Transpose | [[1, 0, 5], [2, 1, 6], [3, 4, 0]] |
| Inverse | [[−24, 18, 5], [20, −15, −4], [−5, 4, 1]] |
You can check the inverse: A × A⁻¹ equals the identity matrix.
How to Use the 3x3 Matrix Calculator
Use the matrix calculator in four steps.
Step 1: Enter the Nine Values of Your Matrix
Type each number into the 3x3 grid, row by row. Leave no cell empty; use 0 where needed.
Step 2: Click Analyze Matrix
Press Analyze Matrix to run all calculations. Use Clear Grid to start again with a new matrix.
Step 3: Read the Determinant and Trace
The top cards show the determinant |A| and the trace tr(A). A determinant of 0 means the matrix has no inverse.
Step 4: Check the Transpose and Inverse
Below, see the transpose Aᵀ and, if it exists, the inverse A⁻¹ written as a 3x3 matrix.
Where 3x3 Matrices Are Used
- Solving systems of three linear equations.
- 3D rotations and transformations in graphics and games.
- Engineering problems with forces, stresses and circuits.
- Statistics, such as covariance matrices.
Why Use a Matrix Calculator?
Finding a 3x3 inverse by hand takes many steps, and one sign error spoils the whole answer. A matrix calculator gives instant, reliable results and lets you focus on understanding the method instead of the arithmetic.
Frequently Asked Questions
Is this matrix calculator free?
Yes. It is free with no sign-up.
Can it find the inverse of any matrix?
Only if the determinant is not zero. A matrix with determinant 0 is singular and has no inverse.
What is the trace of a matrix?
The sum of the numbers on the main diagonal, from top left to bottom right.
Does it support decimals and negatives?
Yes. Enter any real numbers.
Can I use it for 2x2 matrices?
Enter your 2x2 values in the top left and set the third row and column to 0 except a 1 in the bottom right corner.
How do I check my inverse?
Multiply A by A⁻¹; the result should be the identity matrix.