Invertible matrix

Definition

An invertible matrix, also known as a non-singular matrix, is a square matrix that has an inverse. This means that there exists another matrix, called the inverse, which when multiplied with the original matrix results in the identity matrix. The concept of invertible matrices is crucial in linear algebra as it directly relates to the solvability of linear systems and the diagonalization process.

5 Must Know Facts For Your Next Test

1. A square matrix is invertible if and only if its determinant is non-zero.
2. The inverse of an invertible matrix A is denoted as A^{-1}, and it satisfies the equation A * A^{-1} = I, where I is the identity matrix.
3. If a matrix is invertible, its rows and columns are linearly independent.
4. The process of diagonalizing a matrix relies on whether it can be expressed in terms of its eigenvalues and eigenvectors, which are related to its invertibility.
5. For a diagonalizable matrix, if it is also invertible, its eigenvalues must all be non-zero.

Review Questions

• How does the determinant of a matrix relate to its invertibility?
  • The determinant of a square matrix plays a crucial role in determining its invertibility. Specifically, a square matrix is invertible if and only if its determinant is non-zero. If the determinant equals zero, the matrix is singular and does not have an inverse, meaning that certain linear systems associated with it cannot be solved uniquely.
• Discuss how an invertible matrix can be diagonalized and what conditions are necessary for this process.
  • For a square matrix to be diagonalized, it must have enough linearly independent eigenvectors to form a basis for its space. An invertible matrix often meets this requirement since its eigenvalues will generally be non-zero. If the matrix has distinct eigenvalues, it guarantees that it can be diagonalized. Thus, the relationship between invertibility and diagonalizability highlights how structural properties of matrices influence their representation.
• Evaluate the implications of having an invertible matrix in solving linear systems and how this connects to the concept of eigenvalues.
  • An invertible matrix significantly impacts the solutions of linear systems because it guarantees a unique solution for any given set of equations represented by that matrix. This unique solution arises from the existence of an inverse that can be applied to isolate variables. Furthermore, eigenvalues help determine whether a transformation represented by the invertible matrix is stretching or compressing space without collapsing dimensions, reinforcing how these concepts interrelate in analyzing linear transformations.

Related terms

identity matrix: A square matrix with ones on the diagonal and zeros elsewhere, which acts as the multiplicative identity in matrix multiplication.

determinant: A scalar value that can be computed from the elements of a square matrix, which provides important information about the matrix, including whether it is invertible or not.

eigenvalues: Special numbers associated with a matrix that can indicate important properties of the matrix, particularly in relation to its diagonalization.