INVERTIBLE MATRIX Meaning and
Definition
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An invertible matrix, also known as a non-singular matrix or a non-degenerate matrix, refers to a square matrix that possesses an inverse matrix. In linear algebra, a matrix is a rectangular array of numbers arranged in rows and columns. A square matrix has an equal number of rows and columns. The inverse matrix of a given n x n square matrix A is denoted as A⁻¹, and it exists only if the determinant of matrix A is non-zero.
When a matrix is invertible, multiplying it by its inverse matrix yields the identity matrix. The identity matrix, denoted as I, is a square matrix with ones on the main diagonal and zeros elsewhere. It functions as the neutral element for matrix multiplication, similar to how the number one does in regular algebraic operations.
An invertible matrix plays a crucial role in various areas of mathematics, including solving systems of linear equations, determining the rank of a matrix, and finding the eigenvalues and eigenvectors. It also allows the possibility of applying transformational operations on vectors, such as rotations, translations, and scaling.
In practical terms, an invertible matrix represents a linear transformation that can be easily reversed or undone. This property makes invertible matrices invaluable in computer graphics, cryptography, engineering, physics, and numerous other fields where linear operations are employed for calculations, data processing, or modeling complex systems.
Etymology of INVERTIBLE MATRIX
The term "invertible matrix" originates from the combination of two different aspects: "invertible" and "matrix".
1. "Invertible": The word "invertible" comes from the Latin word "invertere", which means "to turn upside down" or "to reverse". In mathematics, an invertible object is one that can be flipped or reversed in some way. In the context of linear algebra, an invertible matrix refers to a square matrix that has an inverse, which means it can be reversed or "turned upside down" by means of matrix multiplication. Another term used interchangeably for an invertible matrix is "non-singular matrix".
2. "Matrix": The term "matrix" has its roots in the Latin word "matrix", which means "womb" or "source". In mathematics, a matrix is a rectangular or square grid of numbers or variables arranged in rows and columns.
