In each of the following, find matrices A, x, and b such...

In each of the following, find matrices $A, x$, and $b$ such that the given system of linear equations can be expressed as the single matrix equation $A x=b$. $\left.\begin{array}{r}2 x_{1}-3 x_{2}+5 x_{3}=7 \\ \text { (a) } 9 x_{1}-x_{2}+x_{3}=-1 \\ x_{1}+5 x_{2}+4 x_{3}=0\end{array}\right\}$

In each of the following, find matrices $A, x$, and $b$ such that the given system of linear equations can be expressed as the single matrix equation $A x=b$.
$\left.\begin{array}{r}2 x_{1}-3 x_{2}+5 x_{3}=7 \\ \text { (a) } 9 x_{1}-x_{2}+x_{3}=-1 \\ x_{1}+5 x_{2}+4 x_{3}=0\end{array}\right\}$

Key Concepts

Matrix Representation of Linear Systems

This concept involves expressing a system of linear equations as a single equation in the form A x = b, where A is the matrix of coefficients, x is the vector of variables, and b is the vector of constants. This form is useful for applying methods of linear algebra to solve the system and for gaining insight into properties like consistency and uniqueness of solutions.

Coefficient Matrix

The coefficient matrix, denoted by A, is constructed by arranging all the coefficients of the variables in the system into a rectangular array. Each row in the matrix represents the coefficients from one equation, and each column corresponds to one variable. This matrix encapsulates the structure of the linear relationships between the variables.

Variable Vector

The variable vector, typically denoted by x, is a column vector that contains all the unknown variables of the system. Writing the variables as a vector allows for compact notation and facilitates operations like matrix multiplication, which are central to many methods used to solve linear systems.

Constant Vector

The constant vector, denoted by b, is formed by listing the constants from the right-hand side of the equations into a column vector. This vector represents the terms that the linear combinations of the variables are equal to, and working with it in vector form helps to maintain consistency in the matrix equation representation.

Matrix Multiplication

Matrix multiplication is the process used to combine the coefficient matrix and the variable vector in the equation A x = b. Each entry in the resulting product corresponds to the sum of products of coefficients and variable values from a row of A and the column vector x, reproducing the left-hand side of each individual equation in the system.

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