Matrix Input Form
Formula Used
Addition and Subtraction
Cij = Aij ± Bij. Both matrices must have identical dimensions.
Matrix Multiplication
Cij = Σ AikBkj. Columns of A must equal rows of B.
Hadamard Product
Cij = AijBij. This is element-by-element multiplication for same-sized matrices.
Scalar Multiplication and Transpose
C = kA and (AT)ij = Aji.
Determinant, Inverse, Trace, and Rank
det(A) is computed through elimination. A-1 is found by Gauss-Jordan elimination.
trace(A) = Σ Aii.
rank(A) equals the number of pivots in row-echelon form.
Variables Inside Entries
Each cell can contain an expression. The calculator substitutes x, y, z, a, b, and c with your values before computing the chosen operation.
How to Use This Calculator
- Choose the matrix operation you want to perform.
- Set the sizes for Matrix A and Matrix B.
- Enter numbers or expressions in each matrix cell.
- Type values for x, y, z, a, b, and c.
- If using scalar multiplication, enter the value for k.
- Press Calculate Matrix Result.
- Review the result above the form, inspect the graph, and export CSV or PDF if needed.
Example Data Table
| Case | Matrix A | Matrix B | Variables | Operation | Expected Output |
|---|---|---|---|---|---|
| Example 1 | [[x, 1], [2, y]] | [[1, z], [3, a]] | x=2, y=3, z=4, a=1 | A + B | [[3, 5], [5, 4]] |
| Example 2 | [[x, 1], [2, y]] | [[1, 0], [z, 3]] | x=2, y=3, z=4 | A × B | [[6, 3], [14, 9]] |
| Example 3 | [[x, 1], [2, y]] | Not required | x=2, y=3 | det(A) | 4 |
FAQs
1. What expressions can I enter inside matrix cells?
You can use numbers, decimals, parentheses, and variables x, y, z, a, b, and c. Supported operators are +, -, *, /, and ^. Examples include 2*x+3, (y-1)/2, and z^2.
2. Why does the inverse sometimes fail?
An inverse exists only for square, non-singular matrices. If the determinant is zero, or the matrix loses pivot positions during elimination, the calculator reports that the inverse does not exist.
3. Can I multiply matrices with different sizes?
Yes, but only when the number of columns in Matrix A equals the number of rows in Matrix B. If that condition fails, the multiplication is undefined and the calculator shows an error.
4. Does the graph use symbolic values directly?
No. The graph is drawn after variables are replaced with your numeric inputs. That means the Plotly output always visualizes evaluated numbers, not raw symbolic text expressions.
5. What does rank tell me?
Rank tells you how many independent rows or columns a matrix has. It is useful for checking linear dependence, solving systems, and understanding whether transformations collapse dimensions.
6. What is exported in the CSV and PDF files?
The export includes the computed result and the summary metrics shown on the page. Matrix results export as table rows, while scalar results export as a labeled value with summary details.
7. Why do some answers show decimals instead of integers?
Operations such as inverse calculation and elimination often produce fractional values. The calculator formats numbers cleanly, but decimals appear whenever the exact result is not a whole number.
8. Which operations require square matrices?
Determinant, inverse, and trace require square matrices. Addition, subtraction, and Hadamard product require equal dimensions. Rank and transpose work on non-square matrices too.