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Matrix Calculator

Perform matrix operations including multiplication and determinants

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Operation

Matrix A

Matrix B

Matrix Calculator Guide: Matrix Operations, Determinants & Linear Algebra

A matrix calculator performs operations such as addition, multiplication, determinant, inverse, transpose, and RREF. It is useful for algebra, graphics, statistics, and systems of equations.

What is Matrix Calculator?

The matrix calculator helps you compute with matrices of chosen size, check dimension compatibility, and get exact or decimal results for determinants, inverses, and products.

How to Use the Matrix Calculator

  1. Choose dimensions (rows x columns) and fill entries.
  2. Select operation: add/subtract, scalar multiply, multiply, determinant, inverse, transpose, or RREF.
  3. Calculate to see the result matrix and key scalars (det, rank).
  4. (Optional) solve Ax = b from RREF or using inverse when defined.
  5. Copy results for homework or engineering notes.

Formulas & Methods

  • Addition/Subtraction (same size): (A +- B)_(i,j) = A(i,j) +- B(i,j)
  • Scalar multiply: (kA)_(i,j) = k * A(i,j)
  • Product: (AB)_(i,j) = sum over k of A(i,k) * B(k,j) (A: m x n, B: n x p)
  • Determinant (2x2): det([[a,b],[c,d]]) = ad - bc
  • Inverse (2x2): A^-1 = (1/(ad - bc))*[[d,-b],[-c,a]] (det != 0)
  • RREF: Gaussian elimination with pivoting to obtain leading 1s and zeros elsewhere.

Assumptions & limitations

  • Floating-point arithmetic introduces rounding; exact fractions may be available for small integers.
  • Non-square matrices have no determinant or inverse.
  • Singular matrices (det = 0) are not invertible.

Examples

Example A — Multiply
A = [[1,2],[3,4]], B = [[2,0],[1,2]]
AB = [[1*2+2*1, 1*0+2*2], [3*2+4*1, 3*0+4*2]] = [[4, 4], [10, 8]].

Example B — Inverse (2x2)
A = [[4,7],[2,6]]det = 4*6 - 7*2 = 10,
A^-1 = (1/10)*[[6,-7],[-2,4]].

| Task | Condition | Output | |---|---|---| | det(A) | A square | scalar | | A^-1 | det(A) != 0 | square | | Ax=b | columns(A)=rows(b) | solution x |

Pro Tips & Best Practices

  • Check dimensions before multiplying; inner sizes must match.
  • Use RREF to spot rank deficiency and inconsistent systems.
  • Scale rows to improve numerical stability.
  • For large problems, prefer LU/QR over explicit inverses.

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FAQ

Q: Which operations are supported?

A: Addition, subtraction, scalar multiply, matrix multiply, determinant, inverse, transpose, and row-reduced echelon form (RREF).

Q: When is a matrix invertible?

A: A square matrix is invertible if its determinant is non-zero and the matrix is full rank.

Q: What sizes can I multiply?

A: If A is m x n and B is n x p, the product AB is m x p; inner dimensions must match.

Q: How is the inverse computed?

A: Typically via Gauss-Jordan elimination or LU decomposition; small 2x2 uses the closed form.

Q: Can I solve linear systems?

A: Yes—enter the augmented matrix and compute RREF or use A^-1 b when A is invertible.

Call to Action

Build your matrices, pick an operation, and compute—use RREF to solve systems or verify homework steps quickly.