An invertible matrix, also known as a non-singular or non-degenerate matrix, is a square matrix that has an inverse, meaning there exists another matrix which, when multiplied with the original, yields the identity matrix. The existence of an inverse is equivalent to the matrix having a non-zero determinant and full rank, ensuring that the linear transformation it represents is bijective.

Invertible Matrix

invertible matrix

A non-singular matrix, also known as an invertible matrix, is a square matrix that has a non-zero determinant and thus possesses an inverse. This property is crucial in solving systems of linear equations, as it ensures the existence of a unique solution when the matrix is used as a coefficient matrix.

Non-singular Matrix

non-degenerate matrix

A square matrix is a matrix with the same number of rows and columns, often used to represent linear transformations in vector spaces. Its properties, such as determinant, eigenvalues, and trace, are foundational in linear algebra and have significant applications in various scientific fields.

square matrix

An identity matrix is a square matrix with ones on the main diagonal and zeros elsewhere, serving as the multiplicative identity in matrix algebra. This means that when any matrix is multiplied by an identity matrix of compatible dimensions, the original matrix is unchanged, analogous to multiplying a number by one in arithmetic.

identity matrix

inverse

The determinant is a scalar value that can be computed from the elements of a square matrix and provides important properties of the matrix, such as whether it is invertible. It is also used in various applications such as solving systems of linear equations, finding volumes in geometry, and analyzing linear transformations.

determinant

A matrix is considered full rank if its rank is equal to the smallest dimension of the matrix, which means all its rows or columns are linearly independent. This property is crucial for solving systems of linear equations and ensures that the matrix has a unique solution or a unique least squares approximation if the system is inconsistent.

full rank

A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication, mapping lines to lines or points through the origin. These transformations can be represented by matrices, making them fundamental in solving systems of linear equations and understanding geometric transformations in higher dimensions.

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