3x3 Matrix Determinant Calculator

A11	A12	A13	A21	A22	A23	A31	A32	A33	Determinant
2	1	3	0	-1	4	5	2	1	17

Formula Used

For a 3×3 matrix

| a b c ; d e f ; g h i |

the determinant is:

det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)

This calculator also derives the minor matrix, cofactor matrix, adjugate, inverse when possible, trace, rank, row sums, and column sums.

FAQs

1. What does the determinant tell me?

The determinant shows whether the matrix is singular, how area or volume scales, and whether an inverse exists. A zero determinant means the matrix collapses dimensions and has no inverse.

2. Why is a zero determinant important?

A zero determinant means the rows or columns are linearly dependent. The matrix cannot be inverted, and it does not preserve full three-dimensional information.

3. What are minors and cofactors?

A minor is the determinant of the 2×2 matrix left after removing one row and one column. A cofactor applies a sign pattern to that minor.

4. When does the inverse exist?

The inverse exists only when the determinant is non-zero. In that case, the calculator divides the adjugate matrix by the determinant to produce the inverse.

5. Why does the calculator show rank?

Rank tells you how many independent directions the matrix preserves. Rank 3 means full dimensional strength, while smaller ranks indicate reduced independence between rows or columns.

6. What does the Plotly graph represent?

The graph visualizes the three first-row Laplace expansion contributions. Adding those contributions gives the final determinant, making the computation easier to interpret.

7. Can I use decimal and negative values?

Yes. The calculator accepts positive numbers, negative numbers, and decimals. It validates numeric input and formats output using your chosen precision setting.

8. Is this useful for study and checking homework?

Yes. It is helpful for learning determinant expansion, verifying hand calculations, understanding singularity, and reviewing minors, cofactors, and inverse-related concepts.