Understanding Matrix Classification

The question asks us to classify the given matrix \(A = \left[ {\begin{array}{} 0&1\\ 1&0 \end{array}} \right]\) based on its properties. We need to determine if it is a Singular, Involuntary, Nilpotent, or Idempotent matrix.

Defining Matrix Types

Let's define each type of matrix mentioned in the options:

Singular Matrix: A square matrix is singular if its determinant is equal to zero. That is, if \(\det(A) = 0\).

Involuntary Matrix: A square matrix \(A\) is involuntary if squaring the matrix results in the identity matrix. That is, if \(A^2 = I\), where \(I\) is the identity matrix of the same order as \(A\).

Nilpotent Matrix: A square matrix \(A\) is nilpotent if there exists some positive integer \(k\) such that \(A^k = 0\), where \(0\) is the zero matrix of the same order as \(A\). The smallest such \(k\) is called the index of nilpotency.

Idempotent Matrix: A square matrix \(A\) is idempotent if squaring the matrix results in the matrix itself. That is, if \(A^2 = A\).

Analyzing Matrix A

We are given the matrix \(A = \left[ {\begin{array}{} 0&1\\ 1&0 \end{array}} \right]\). This is a \(2 \times 2\) square matrix.

Check if A is Singular

To check if A is singular, we calculate its determinant:

\(\det(A) = \det\left[ {\begin{array}{} 0&1\\ 1&0 \end{array}} \right]\)

\(\det(A) = (0 \times 0) - (1 \times 1) = 0 - 1 = -1\)

Since \(\det(A) = -1 \neq 0\), the matrix A is not a singular matrix.

Check if A is Involuntary or Idempotent

To check if A is involuntary or idempotent, we need to calculate \(A^2\):

\(A^2 = A \times A = \left[ {\begin{array}{} 0&1\\ 1&0 \end{array}} \right] \times \left[ {\begin{array}{} 0&1\\ 1&0 \end{array}} \right]\)

Multiplying the matrices:

\(A^2 = \left[ {\begin{array}{} (0 \times 0 + 1 \times 1)&(0 \times 1 + 1 \times 0)\\ (1 \times 0 + 0 \times 1)&(1 \times 1 + 0 \times 0) \end{array}} \right]\)

\(A^2 = \left[ {\begin{array}{} (0 + 1)&(0 + 0)\\ (0 + 0)&(1 + 0) \end{array}} \right]\)

\(A^2 = \left[ {\begin{array}{} 1&0\\ 0&1 \end{array}} \right]\)

The resulting matrix \(\left[ {\begin{array}{} 1&0\\ 0&1 \end{array}} \right]\) is the \(2 \times 2\) identity matrix, denoted as \(I_2\).

So, we have \(A^2 = I_2\).

Comparing \(A^2\) with A: \(A^2 = \left[ {\begin{array}{} 1&0\\ 0&1 \end{array}} \right]\) and \(A = \left[ {\begin{array}{} 0&1\\ 1&0 \end{array}} \right]\). Since \(A^2 \neq A\), the matrix A is not an idempotent matrix.

Comparing \(A^2\) with I: \(A^2 = \left[ {\begin{array}{} 1&0\\ 0&1 \end{array}} \right]\) and \(I_2 = \left[ {\begin{array}{} 1&0\\ 0&1 \end{array}} \right]\). Since \(A^2 = I\), the matrix A is an involuntary matrix.

Check if A is Nilpotent

We found that \(A^2 = I\). Let's calculate higher powers of A:

\(A^3 = A^2 \times A = I \times A = A = \left[ {\begin{array}{} 0&1\\ 1&0 \end{array}} \right]\)

\(A^4 = A^2 \times A^2 = I \times I = I = \left[ {\begin{array}{} 1&0\\ 0&1 \end{array}} \right]\)

The powers of A alternate between A and I. The matrix \(A^k\) will never be the zero matrix \(\left[ {\begin{array}{} 0&0\\ 0&0 \end{array}} \right]\) for any positive integer \(k\). Therefore, the matrix A is not a nilpotent matrix.

Conclusion

Based on our analysis, the matrix \(A = \left[ {\begin{array}{} 0&1\\ 1&0 \end{array}} \right]\) satisfies the condition \(A^2 = I\). By definition, a matrix satisfying this property is called an involuntary matrix.

Therefore, the matrix A is an involuntary matrix.

Revision Table: Types of Matrices

Additional Information on Matrix Properties

Understanding matrix properties like singularity, involuntariness, nilpotency, and idempotency is crucial in linear algebra. These properties help classify matrices and predict their behavior in various operations.

An involuntary matrix is also its own inverse, because if \(A^2 = I\), then \(A \times A = I\), which implies \(A^{-1} = A\). The matrix \(A = \left[ {\begin{array}{} 0&1\\ 1&0 \end{array}} \right]\) is a symmetric matrix that represents a reflection across the line \(y=x\) in 2D space. Performing the reflection twice returns the original point, which aligns with the property \(A^2 = I\).

Idempotent matrices represent projections. For example, projecting a vector onto the x-axis.

Nilpotent matrices are related to transformations that eventually collapse the space to the zero vector.

Singular matrices indicate that the linear transformation represented by the matrix collapses dimensions, meaning different non-zero vectors can be mapped to the same vector, or the determinant is zero. Non-singular matrices are invertible.

Being able to quickly check these properties by calculating the determinant, \(A^2\), or higher powers helps in solving problems involving matrix transformations and systems of linear equations.