Question

Is it possible for a matrix to be its own inverse?

Short answer

Yes, a matrix can be its own inverse, such as identity, or permutation matrices.

Step-by-step solution

Define What It Means to Be an Inverse

A matrix \( A \) is said to be invertible if there exists another matrix \( B \) such that \( AB = BA = I \), where \( I \) is the identity matrix. If a matrix is its own inverse, it implies \( A = B \) and thus \( A^2 = I \).

Set the Equation for Self-Inverse Matrices

For a matrix \( A \) to be its own inverse, the condition required is \( A^2 = I \). This means that when the matrix is multiplied by itself, the result must be the identity matrix.

Check for Conditions in 2x2 matrices

Consider a 2x2 matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \). If \( A \) is its own inverse, \( A^2 = I \), leading to the equations: \( a^2 + bc = 1 \), \( ab + bd = 0 \) \( ac + cd = 0 \), \( bc + d^2 = 1 \).

Verify Existence of Solutions

Check if such a matrix \( A \) exists where all these conditions are satisfied. Possible matrices include ones like \( A = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \) or \( A = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} \), as they satisfy the equations derived in the previous steps.

Key concepts

Identity Matrix

An identity matrix is a special type of matrix that acts like the number 1 does in multiplication. It's denoted by the symbol \( I \) and has distinctive characteristics that make it stand out. For any square matrix \( A \) of size \( n \times n \), multiplying \( A \) by an identity matrix doesn't change \( A \). This is similar to how multiplying by 1 leaves a number unchanged.

An identity matrix of size \( n \times n \) is a square array with ones on the diagonal (from the top left to the bottom right) and zeros everywhere else. For example, the 2x2 identity matrix looks like this:

• \( \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \)

Think of the identity matrix as the baseline or the neutral element in matrix operations, ensuring that the original matrix remains unaltered when multiplied by it.

Invertible Matrix

An invertible matrix, also known as a non-singular or non-degenerate matrix, is one that has an inverse. This means there's another matrix that, when multiplied with the original matrix, results in the identity matrix.

• If matrix \( A \) is an invertible matrix, there exists a matrix \( B \) such that \( A \times B = I \), where \( I \) is the identity matrix.

For a matrix to be invertible, it must be a square matrix (same number of rows and columns) and must also have a non-zero determinant. The ability to find an inverse matrix is crucial in solving systems of linear equations, as it allows for the isolation of variables and solving for unknowns.

Self-Inverse Matrices

Self-inverse matrices, or involutory matrices, are matrices that serve as their own inverses. In simpler terms, if you multiply a self-inverse matrix by itself, you get the identity matrix.

Essentially, this means if a matrix \( A \) is self-inverse, then \( A^2 = I \). These matrices have unique properties and an interesting geometric interpretation - they often correspond to reflections across certain lines or planes. For example:

• The 2x2 matrix \( \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} \) is self-inverse because multiplying it by itself results in the identity matrix.

This peculiar property is leveraged in various applications, particularly in computer graphics and cryptographic algorithms.

2x2 Matrices

2x2 matrices are amongst the simplest forms of matrices and provide a foundational block for understanding matrix operations. They are square matrices with two rows and two columns, often represented as:

• \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \)

Despite their simplicity, 2x2 matrices exhibit intriguing behaviors. They can be used to perform linear transformations such as rotations, reflections, and shears in a 2D plane.

When working with 2x2 matrices, it is also straightforward to compute the determinant, which is \( ad - bc \), a key factor in determining whether a matrix is invertible:

• If the determinant is not zero, the matrix has an inverse.
• If it equals zero, the matrix does not have an inverse and is called singular.

These matrices offer a compact way to explore and apply the principles of linear algebra in various mathematical and practical scenarios.