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Matrix Determinant Calculator

Evaluate a real three-by-three determinant and check whether the matrix is singular. Enter the elements row by row, following the row and column labels. The scalar output summarizes an important property of the matrix rather than giving its inverse. In a linear transformation, the determinant represents signed volume scaling. To evaluate a two-by-two determinant, place that matrix in the upper-left block and enter zeros elsewhere except a 1 in row 3, column 3.

Enter every field, including explicit zeros. Decimal and scientific notation are accepted; commas, fractions and blank fields are rejected. Each input must be finite and no larger than 1 billion in absolute value.

3 × 3 determinant6

How to use this calculator

  1. Match each input to its labeled coordinate, coefficient or matrix position.
  2. Check the signs and use one consistent coordinate convention.
  3. Select Calculate, or press Enter, and read the labeled results.

Formula and method

det(A) = a(ei − fh) − b(di − fg) + c(dh − eg), for rows (a,b,c), (d,e,f), (g,h,i).

The calculation runs locally in your browser. Changing an input requires selecting Calculate again. If an input or mathematical condition is invalid, the previous result is hidden instead of leaving a stale answer visible.

Worked example

The diagonal matrix with rows (1,0,0), (0,2,0) and (0,0,3) has determinant 1×2×3 = 6.

Limitations and interpretation

Only the stated three-by-three layout is evaluated. A result of zero indicates singularity mathematically, but decimal rounding can obscure near-singular cases. This is floating-point calculation, not symbolic algebra; cancellation can reduce accuracy for nearly dependent rows.

Results are displayed to approximately 10 significant digits. The underlying calculation uses ordinary floating-point numbers and is not exact symbolic arithmetic. Rounded results can differ slightly from hand calculations. For a measured quantity, choose precision that reflects the measurements rather than all the digits shown. Verify consequential calculations independently.

Frequently asked questions

How can I enter a 2 × 2 matrix?

Use rows (a,b,0), (c,d,0), (0,0,1). Its 3 × 3 determinant equals ad − bc.

Can I enter negative or decimal values?

Yes, provided the values meet the mathematical conditions described above. Type a normal hyphen for a negative sign. Do not type units, commas or a fraction such as 1/2; enter 0.5 instead.

Why did the calculator hide my previous result?

A missing or invalid input, undefined operation or unstable numerical case produces an explanation. Correct the input and calculate again to obtain a result for the current values.

Review notes and reference

Reviewed October 3, 2026. The stated default example and a boundary or algebraic check were tested against expected values. Input validation checks included blank fields, nonnumeric text and values outside the supported range. Wolfram MathWorld formula reference provides mathematical background. This tool is for calculation and education; its output alone does not validate a professional design.

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