Calculator
Choose a square matrix size, enter values for both matrices, and calculate the determinant of their sum.
Example Data Table
This sample shows a completed 3 × 3 setup and the resulting determinant after matrix addition.
| Size | Matrix A | Matrix B | Matrix A + B | det(A + B) |
|---|---|---|---|---|
| 3 × 3 | [[2, 1, 0], [1, 3, 2], [4, 0, 1]] | [[1, 0, 2], [0, 1, -1], [3, 2, 1]] | [[3, 1, 2], [1, 4, 1], [7, 2, 2]] | -29 |
Formula Used
First, the calculator adds the two matrices entry by entry:
C = A + B
For each position, the summed entry is:
cᵢⱼ = aᵢⱼ + bᵢⱼ
Then the calculator finds the determinant of the summed matrix:
det(C) = det(A + B)
For a 2 × 2 matrix, the determinant is:
det(C) = c₁₁c₂₂ − c₁₂c₂₁
For larger matrices, the page uses Gaussian elimination. Row swaps change the sign, and the product of pivot values gives the determinant.
Important: determinants are not additive in general,
so det(A + B) usually differs from det(A) + det(B).
How to Use This Calculator
- Select a square matrix size from 2 × 2 up to 5 × 5.
- Enter the values of Matrix A and Matrix B.
- Choose how many decimal places you want in the output.
- Press Calculate Determinant of A + B.
- Review the result panel above the form for determinants, matrix tables, and the chart.
- Use the CSV or PDF buttons to export the completed report.
Frequently Asked Questions
1. What does this calculator compute?
It adds two square matrices and calculates the determinant of the resulting matrix. It also shows Matrix A, Matrix B, Matrix A + B, trace, Frobenius norm, and a comparison chart.
2. Why must both matrices be square?
A determinant exists only for square matrices. That means the number of rows and columns must match, and both matrices must share the same dimensions before addition.
3. Is det(A + B) the same as det(A) + det(B)?
No. Determinants do not distribute over matrix addition in general. This is why the calculator first forms the sum matrix and then computes its determinant separately.
4. What happens if the summed matrix is singular?
If the determinant of the summed matrix equals zero, the matrix is singular. That means it does not have an inverse, and the result panel labels it clearly.
5. How are larger determinants calculated?
The calculator uses Gaussian elimination for larger matrices. This method is efficient, reduces the matrix step by step, and preserves the determinant relationship after accounting for row swaps.
6. Can I use decimals and negative numbers?
Yes. The input fields accept integers, decimals, and negative values. You can also choose the number of decimal places used when displaying the final results.
7. What do the CSV and PDF exports include?
Both exports include the matrix size, Matrix A, Matrix B, Matrix A + B, and the main summary metrics. They are useful for reports, notes, or coursework records.
8. Does the graph change with my inputs?
Yes. After every calculation, the graph updates to compare the determinants of Matrix A, Matrix B, and the summed matrix using your current values.