Select The Correct Answer:If A 2 = A A^2 = A A 2 = A , Which Matrix Satisfies This Condition?A. $\left[\begin{array}{cc}5 & 5 \ -4 & -4\end{array}\right]B. $\left[\begin{array}{cc}6 & 5 \ 5 & 6\end{array}\right]C.

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Introduction

In linear algebra, a matrix that satisfies the condition A2=AA^2 = A is known as an idempotent matrix. This property is crucial in various applications, including signal processing, image analysis, and machine learning. In this article, we will explore the concept of idempotent matrices and determine which of the given matrices satisfies the condition A2=AA^2 = A.

What is an Idempotent Matrix?

An idempotent matrix is a square matrix that, when multiplied by itself, results in the same matrix. Mathematically, this can be represented as A2=AA^2 = A, where AA is the idempotent matrix. This property is essential in various applications, including signal processing, image analysis, and machine learning.

Properties of Idempotent Matrices

Idempotent matrices have several interesting properties. Some of the key properties include:

  • Diagonalization: Idempotent matrices can be diagonalized, meaning that they can be transformed into a diagonal matrix using a similarity transformation.
  • Eigenvalues: The eigenvalues of an idempotent matrix are either 0 or 1.
  • Rank: The rank of an idempotent matrix is equal to the number of non-zero eigenvalues.

How to Determine if a Matrix is Idempotent

To determine if a matrix is idempotent, we can use the following steps:

  1. Square the matrix: Multiply the matrix by itself to obtain the square of the matrix.
  2. Compare the result with the original matrix: If the square of the matrix is equal to the original matrix, then the matrix is idempotent.

Example Matrices

Let's consider the three matrices given in the problem:

A. [55−4−4]\left[\begin{array}{cc}5 & 5 \\ -4 & -4\end{array}\right] B. [6556]\left[\begin{array}{cc}6 & 5 \\ 5 & 6\end{array}\right] C. [1000]\left[\begin{array}{cc}1 & 0 \\ 0 & 0\end{array}\right]

We will square each of these matrices and compare the result with the original matrix to determine if it is idempotent.

Matrix A

To square Matrix A, we multiply it by itself:

[55−4−4]×[55−4−4]=[25−2025−20−20+16−20+16]=[55−4−4]\left[\begin{array}{cc}5 & 5 \\ -4 & -4\end{array}\right] \times \left[\begin{array}{cc}5 & 5 \\ -4 & -4\end{array}\right] = \left[\begin{array}{cc}25-20 & 25-20 \\ -20+16 & -20+16\end{array}\right] = \left[\begin{array}{cc}5 & 5 \\ -4 & -4\end{array}\right]

Since the square of Matrix A is equal to the original matrix, Matrix A is idempotent.

Matrix B

To square Matrix B, we multiply it by itself:

[6556]×[6556]=[36+2536+3030+3625+36]=[61666661]\left[\begin{array}{cc}6 & 5 \\ 5 & 6\end{array}\right] \times \left[\begin{array}{cc}6 & 5 \\ 5 & 6\end{array}\right] = \left[\begin{array}{cc}36+25 & 36+30 \\ 30+36 & 25+36\end{array}\right] = \left[\begin{array}{cc}61 & 66 \\ 66 & 61\end{array}\right]

Since the square of Matrix B is not equal to the original matrix, Matrix B is not idempotent.

Matrix C

To square Matrix C, we multiply it by itself:

[1000]×[1000]=[1000]\left[\begin{array}{cc}1 & 0 \\ 0 & 0\end{array}\right] \times \left[\begin{array}{cc}1 & 0 \\ 0 & 0\end{array}\right] = \left[\begin{array}{cc}1 & 0 \\ 0 & 0\end{array}\right]

Since the square of Matrix C is equal to the original matrix, Matrix C is idempotent.

Conclusion

In this article, we explored the concept of idempotent matrices and determined which of the given matrices satisfies the condition A2=AA^2 = A. We found that Matrix A and Matrix C are idempotent, while Matrix B is not. The properties of idempotent matrices, including diagonalization, eigenvalues, and rank, were also discussed. By following the steps outlined in this article, you can determine if a matrix is idempotent and apply this knowledge to various applications in linear algebra and beyond.

References

  • Horn, R. A., & Johnson, C. R. (2013). Matrix Analysis. Cambridge University Press.
  • Golub, G. H., & Van Loan, C. F. (2013). Matrix Computations. Johns Hopkins University Press.
  • Strang, G. (2016). Linear Algebra and Its Applications. Cengage Learning.

Introduction

In our previous article, we explored the concept of idempotent matrices and determined which of the given matrices satisfies the condition A2=AA^2 = A. In this article, we will answer some frequently asked questions about idempotent matrices.

Q1: What is the difference between an idempotent matrix and a singular matrix?

A1: An idempotent matrix is a square matrix that, when multiplied by itself, results in the same matrix. A singular matrix, on the other hand, is a square matrix that has a determinant of zero, meaning that it is not invertible. While an idempotent matrix may be singular, not all singular matrices are idempotent.

Q2: Can an idempotent matrix be invertible?

A2: Yes, an idempotent matrix can be invertible. In fact, if an idempotent matrix is invertible, then its inverse is also idempotent.

Q3: How do I determine if a matrix is idempotent?

A3: To determine if a matrix is idempotent, you can follow these steps:

  1. Square the matrix: Multiply the matrix by itself to obtain the square of the matrix.
  2. Compare the result with the original matrix: If the square of the matrix is equal to the original matrix, then the matrix is idempotent.

Q4: What are some applications of idempotent matrices?

A4: Idempotent matrices have several applications in linear algebra and beyond. Some examples include:

  • Signal processing: Idempotent matrices are used in signal processing to filter out noise and extract relevant information from signals.
  • Image analysis: Idempotent matrices are used in image analysis to segment images and extract features.
  • Machine learning: Idempotent matrices are used in machine learning to regularize models and prevent overfitting.

Q5: Can an idempotent matrix have non-zero eigenvalues?

A5: Yes, an idempotent matrix can have non-zero eigenvalues. In fact, the eigenvalues of an idempotent matrix are either 0 or 1.

Q6: How do I find the eigenvalues of an idempotent matrix?

A6: To find the eigenvalues of an idempotent matrix, you can use the following steps:

  1. Diagonalize the matrix: Diagonalize the idempotent matrix using a similarity transformation.
  2. Extract the eigenvalues: The eigenvalues of the idempotent matrix are the diagonal entries of the diagonalized matrix.

Q7: Can an idempotent matrix be a symmetric matrix?

A7: Yes, an idempotent matrix can be a symmetric matrix. In fact, if an idempotent matrix is symmetric, then its transpose is also idempotent.

Q8: How do I determine if a matrix is idempotent and symmetric?

A8: To determine if a matrix is idempotent and symmetric, you can follow these steps:

  1. Check if the matrix is idempotent: Square the matrix and compare the result with the original matrix.
  2. Check if the matrix is symmetric: Check if the matrix is equal to its transpose.

Conclusion

In this article, we answered some frequently asked questions about idempotent matrices. We discussed the difference between an idempotent matrix and a singular matrix, the properties of idempotent matrices, and some applications of idempotent matrices. We also provided some tips and tricks for working with idempotent matrices.

References

  • Horn, R. A., & Johnson, C. R. (2013). Matrix Analysis. Cambridge University Press.
  • Golub, G. H., & Van Loan, C. F. (2013). Matrix Computations. Johns Hopkins University Press.
  • Strang, G. (2016). Linear Algebra and Its Applications. Cengage Learning.