Introduction to Matrices

What are Matrices?

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. It is a fundamental object in linear algebra and appears in many areas of mathematics and applied sciences.

⎡a₁₁ a₁₂ … a₁n⎤
⎢a₂₁ a₂₂ … a₂n⎥
⎢ ⋮   ⋮   ⋱ ⋮ ⎥
⎣am₁ am₂ … amn⎦

Basic Operations

• Addition: Element‑wise addition of two matrices of the same dimensions.
• Scalar Multiplication: Multiply each entry by a constant.
• Matrix Multiplication: The dot product of rows of the first matrix with columns of the second.

Example of matrix multiplication:

// Multiply A (2×3) by B (3×2)
A = [[1,2,3],
     [4,5,6]];
B = [[7,8],
     [9,10],
     [11,12]];
// Result C = A·B (2×2)
C = [[58, 64],
     [139,154]];

Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix. For a 2×2 matrix:

det([[a,b],
     [c,d]]) = a·d - b·c

It provides information about the matrix such as invertibility.

Inverse Matrix

A matrix A has an inverse A⁻¹ if and only if it is square and its determinant is non‑zero. For a 2×2 matrix:

A⁻¹ = (1/det(A)) * [[d, -b],
                       [-c, a]]

Try the interactive calculator below:

Applications

Matrices are used in:

• Computer graphics (transformations)
• Systems of linear equations
• Markov chains
• Machine learning (weight matrices)