text: # The Mysterious Rank (of a Matrix) Demystified
  
  Let us start with the following system of linear equations (written in matrix form):
  $$AX = B\text{ where } A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix},\, X = \begin{bmatrix} x_{1} \\ x_{2} \\ \cdots \\ x_{n} \end{bmatrix},\, B = \begin{bmatrix} b_{1} \\ b_{2} \\ \cdots \\ b_{m} \end{bmatrix}$$ The augmented matrix of the system is: $$\tilde{A} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} & b_{1} \\ a_{21} & a_{22} & \cdots & a_{2n} & b_{2} \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_{m} \end{bmatrix}$$
  
  # The Mysterious Rank (of a Matrix): Elementary Row Operations
  
  In the previous post we formed notions of matrix, determinants, rank of a matrix and system of linear equations. We also mentioned the fundamental theorem on solution of system of linear equations which tells us the importance of rank in deciding the nature of solution of such systems of equations. It is now time to establish the theorem and to do that we start by defining a systematic method of solving such systems of linear equations.
  
  # The Mysterious Rank (of a Matrix)
  
  In this post I will be talking about the simple and beautiful concept of matrices. In particular I will discuss about the rank of a matrix and its usage in solving system of linear equations. In order to keep the post to a reasonable length I will not dwell upon the usual operations on matrices and assume that the reader is familiar with them, but a paragraph or two about these operations definitely makes sense.
explanation: The document is an excerpt from a series of posts about matrices and linear algebra, specifically focusing on the concept of the rank of a matrix and its application in solving systems of linear equations. It introduces the matrix form of a system of linear equations and the augmented matrix. The text does not contain a specific question or a question and answer.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
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distilabel_metadata.role: system
distilabel_metadata.content: # The Mysterious Rank (of a Matrix) Demystified
  
  Let us start with the following system of linear equations (written in matrix form):
  $$AX = B\text{ where } A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \cdots & \cdots & \cdots & \cdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix},\, X = \begin{bmatrix} x_{1} \\ x_{2} \\ \cdots \\ x_{n} \end{bmatrix},\, B = \begin{bmatrix} b_{1} \\ b_{2} \\ \cdots \\ b_{m} \end{bmatrix}$$ The augmented matrix of the system is: $$\tilde{A} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} & b_{1} \\ a_{21} & a_{22} & \cdots & a_{2n} & b_{2} \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_{m} \end{bmatrix}$$
  
  # The Mysterious Rank (of a Matrix): Elementary Row Operations
  
  In the previous post we formed notions of matrix, determinants, rank of a matrix and system of linear equations. We also mentioned the fundamental theorem on solution of system of linear equations which tells us the importance of rank in deciding the nature of solution of such systems of equations. It is now time to establish the theorem and to do that we start by defining a systematic method of solving such systems of linear equations.
  
  # The Mysterious Rank (of a Matrix)
  
  In this post I will be talking about the simple and beautiful concept of matrices. In particular I will discuss about the rank of a matrix and its usage in solving system of linear equations. In order to keep the post to a reasonable length I will not dwell upon the usual operations on matrices and assume that the reader is familiar with them, but a paragraph or two about these operations definitely makes sense.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is an excerpt from a series of posts about matrices and linear algebra, specifically focusing on the concept of the rank of a matrix and its application in solving systems of linear equations. It introduces the matrix form of a system of linear equations and the augmented matrix. The text does not contain a specific question or a question and answer.
  
  Label: no-question