Matrices: the basics

In short: A matrix is a rectangular array of numbers arranged in rows and columns, used to compactly represent multiple equations or datasets. Matrices are a central tool of linear algebra and appear in many fields, from computer graphics to economics.

In more detail: A matrix’s size is given by its number of rows and columns, for example a 3x2 matrix with three rows and two columns.

In detail

• Matrix addition: Two matrices of the same size are added by adding up their corresponding entries.
• Matrix multiplication: When multiplying two matrices, rows of the first are combined with columns of the second, which requires the number of columns in the first to match the number of rows in the second.
• Identity matrix: A square matrix with ones on the diagonal and zeros elsewhere acts like the number 1 in multiplication.
• Systems of linear equations: Matrices neatly summarize multiple equations and allow systematic solution methods like Gaussian elimination.
• Determinant: A single number derived from a square matrix indicates whether the associated system of equations has a unique solution.
• Transposed matrix: Transposing swaps rows and columns, which plays an important role in many calculations.
• Applications: Matrices are used in computer graphics for transformations, in economics for input-output models, and in statistics for data analysis (vektoren-grundlagen).