3×3 Matrix Operations Calculator
Resulting Matrix
Understanding Matrix Calculations
A matrix is a rectangular array of numbers arranged in rows and columns. In mathematics and linear algebra, matrix operations are fundamental for solving systems of linear equations, computer graphics, and data science. This calculator handles 3×3 matrices, which are standard in three-dimensional physics and geometry.
Matrix Addition and Subtraction
To add or subtract two matrices, they must have the same dimensions. You simply add or subtract the corresponding elements in each position. For example, the element at row 1, column 1 of Matrix A is added to the element at row 1, column 1 of Matrix B.
[1, 2] + [3, 4] = [(1+3), (2+4)] = [4, 6]
Matrix Multiplication (The Dot Product)
Matrix multiplication is more complex than simple addition. It is NOT performed element-wise. Instead, you calculate the dot product of the rows of the first matrix and the columns of the second matrix. For a 3×3 matrix, the element at row i and column j in the result is the sum of the products of the elements in row i of A and column j of B.
Key Rules of Matrix Algebra
- Commutativity: Matrix addition is commutative (A + B = B + A), but matrix multiplication is not (A × B ≠ B × A).
- Identity Matrix: A special square matrix with ones on the main diagonal and zeros elsewhere. Multiplying any matrix by the identity matrix results in the original matrix.
- Dimensions: For multiplication to be possible, the number of columns in the first matrix must equal the number of rows in the second matrix.
Real-World Applications
Matrices are used in daily technology more than you might think. Digital images are essentially large matrices of pixel values. When you rotate a photo on your phone, the software performs matrix multiplication on the coordinates of every pixel. In engineering, matrices help calculate the stresses on a bridge or the flow of electricity in a complex circuit.