Matrix Calculator
What is a Matrix?
A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are fundamental tools in mathematics, used to represent linear transformations, solve systems of linear equations, and handle data in computer science and physics. A matrix with m rows and n columns is called an m × n matrix.
What This Calculator Does
This calculator allows you to perform essential matrix operations quickly and accurately. You can:
- Add and Subtract matrices of the same size.
- Multiply matrices (dot product).
- Calculate the Determinant of a square matrix.
- Find the Rank to see the number of linearly independent rows.
- Compute the Inverse of a square matrix (if it exists).
How to Calculate Manually
Matrix Addition and Subtraction
To add or subtract two matrices, you simply add or subtract their corresponding elements. The matrices must have the same dimensions. Given two matrices A and B, the rule is (A ± B)ij = Aij ± Bij.
Example:
Matrix Multiplication
Matrix multiplication is more complex. The element in row i and column j of the result is the "dot product" of row i from the first matrix and column j from the second: (AB)ij = Σₖ Aik Bkj.
Example (2 × 2):
Determinant
The determinant is a special number calculated from a square matrix. For a 2 × 2 matrix, the formula is simple: det( [ [a, b], [c, d] ] ) = ad − bc.
For larger matrices, the calculation involves breaking the matrix down into smaller parts (minors).
Inverse Matrix
The inverse of a matrix A is denoted as A⁻¹. When multiplied by the original matrix, it yields the identity matrix I, so A · A⁻¹ = I.
For a 2 × 2 matrix, the inverse is calculated as A⁻¹ = 1 / det(A) · [ [d, −b], [−c, a] ]. Note: A matrix has an inverse only if its determinant is not zero.