Question

If \(\lambda\) is an eigen value of a non-eingular matrix \(A\), then the eigen value of \(A^{-1}\) is

Short answer

The eigenvalue of \(A^{-1}\) is \(\frac{1}{\lambda}\).

Step-by-step solution

Identify the Relationship Between Matrix A and Its Inverse

For a matrix \(A\), if \(\lambda\) is an eigenvalue, then there exists a non-zero vector \(\mathbf{v}\) such that \(A\mathbf{v} = \lambda \mathbf{v}\). Since \(A\) is invertible (non-singular), \(A^{-1}\) exists.

Apply the Inverse Relationship

Using the equation \(A\mathbf{v} = \lambda \mathbf{v}\), multiply both sides by \(A^{-1}\) to obtain \(\mathbf{v} = \lambda A^{-1}\mathbf{v}\).

Rearrange to Isolate the Inverse Matrix Eigenvalue

Rearrange \(\mathbf{v} = \lambda A^{-1}\mathbf{v}\) to form \(A^{-1}\mathbf{v} = \frac{1}{\lambda}\mathbf{v}\). This implies that \(\frac{1}{\lambda}\) is an eigenvalue of \(A^{-1}\).

Key concepts

Non-singular matrix

A non-singular matrix, often referred to as an invertible or non-degenerate matrix, is a square matrix that has an inverse. This means that for a matrix \(A\), there exists another matrix \(B\) such that multiplying \(A\) by \(B\) results in the identity matrix. Formally, this can be expressed as \(AB = I\) and \(BA = I\), where \(I\) is the identity matrix.

In simpler terms:

• A non-singular matrix is one that can be reversed or "undone" through multiplication with its inverse.
• If a matrix is non-singular, it has a determinant that is not equal to zero. This is a key characteristic since a matrix with a zero determinant is singular, and thus, does not have an inverse.
• Non-singular matrices are crucial in solving systems of linear equations, as they guarantee that a unique solution exists.

Understanding non-singular matrices helps in appreciating their role in matrix theory and their applications in areas like computer graphics and data transformations.

Inverse matrix

The inverse of a matrix \(A\) is denoted as \(A^{-1}\). It serves as a critical concept in linear algebra, particularly when dealing with matrix equations. The inverse is defined so that when it is multiplied by the original matrix, the result is the identity matrix. Mathematically, this relationship can be shown as either \(A \cdot A^{-1} = I\) or \(A^{-1} \cdot A = I\).

Some key points about inverse matrices include:

• Only non-singular matrices, or matrices with a non-zero determinant, have inverses. This means non-invertible matrices lack an inverse.
• The process of finding an inverse can involve various methods, such as row reduction or applying the formula involving the matrix's determinant and adjugate.
• In practical applications, inverse matrices are used in solving linear systems via matrix equations, transforming geometric entities, and cryptography.

Grasping the concept of inverse matrices is essential for understanding how systems of equations can be manipulated and solved efficiently.

Linear algebra

Linear algebra is a branch of mathematics focusing on vector spaces and linear mappings between these spaces. It involves studying matrices, vectors, determinants, and eigenvalues, among other concepts. Linear algebra acts as the backbone of many scientific computations and is widely used in various fields such as engineering, physics, and computer science.

Key areas within linear algebra include:

• Vector Spaces: These are collections of vectors where vector addition and scalar multiplication are defined. They provide the setting for linear equations and transformations.
• Matrices and Determinants: Matrices serve as tools for representing and solving linear equations, while determinants provide important properties of matrices, such as invertibility.
• Eigenvalues and Eigenvectors: These are crucial in understanding matrix behaviors and are particularly important in dynamic systems and stability analysis.

Linear algebra is vital for analyzing and solving linear systems and aids in understanding complex data structures found in disciplines like computer science, machine learning, and data analytics.