Given A and b in Exercises 11 and 12, write the augmented...

Given $A$ and $b$ in Exercises 11 and $12,$ write the augmented matrix for the linear system that corresponds to the matrix equation $A \mathbf{x}=\mathbf{b} .$ Then solve the system and write the solution as a vector. $$ A=\left[\begin{array}{rrr}{1} & {2} & {4} \\ {0} & {1} & {5} \\ {-2} & {-4} & {-3}\end{array}\right], \mathbf{b}=\left[\begin{array}{r}{-2} \\ {2} \\ {9}\end{array}\right] $$

Given $A$ and $b$ in Exercises 11 and $12,$ write the augmented matrix for the linear system that corresponds to the matrix equation $A \mathbf{x}=\mathbf{b} .$ Then solve the system and write the solution as a vector.
$$
A=\left[\begin{array}{rrr}{1} & {2} & {4} \\ {0} & {1} & {5} \\ {-2} & {-4} & {-3}\end{array}\right], \mathbf{b}=\left[\begin{array}{r}{-2} \\ {2} \\ {9}\end{array}\right]
$$

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Key Concepts

Gaussian Elimination

Gaussian elimination is a method for solving systems of linear equations. It involves applying a sequence of elementary row operations to the augmented matrix to reduce it to a row-echelon or reduced row-echelon form, from which the solutions can be obtained easily.

Solution Vector

The solution vector is the result of solving the system of linear equations, representing the values of the variables that satisfy all the equations simultaneously. It is typically written as a column vector, corresponding to the order of variables in the original equation.

Augmented Matrix

An augmented matrix is constructed by appending the constant vector b to the coefficient matrix A, thereby combining the entire system into one matrix. This representation facilitates the use of systematic row reduction methods to find the solutions of the system.

Matrix Equation

A matrix equation of the form A x = b is a compact way of representing a system of linear equations, where A is a matrix of coefficients, x is the vector of variables, and b is the vector of constants. This formulation allows the application of matrix operations to study and solve the system.

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