# 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.
The text passage introduces the concept of matrices and the rank of a matrix, particularly in the context of solving systems of linear equations. It provides the matrix form of a system of linear equations and the augmented matrix, but does not contain any specific mathematical questions or answers.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 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.
user
The text passage introduces the concept of matrices and the rank of a matrix, particularly in the context of solving systems of linear equations. It provides the matrix form of a system of linear equations and the augmented matrix, but does not contain any specific mathematical questions or answers.

TYPE: NO_QUESTION