In Exercises 33–36, write each matrix equation as a system...

In Exercises 33–36, write each matrix equation as a system of linear equations without matrices. $$ \left[\begin{array}{rrr}{-1} & {0} & {1} \\ {0} & {-1} & {0} \\ {0} & {1} & {1}\end{array}\right]\left[\begin{array}{l}{x} \\ {y} \\ {z}\end{array}\right]=\left[\begin{array}{r}{-4} \\ {2} \\ {4}\end{array}\right] $$

In Exercises 33–36, write each matrix equation as a system of linear equations without matrices.
$$
\left[\begin{array}{rrr}{-1} & {0} & {1} \\ {0} & {-1} & {0} \\ {0} & {1} & {1}\end{array}\right]\left[\begin{array}{l}{x} \\ {y} \\ {z}\end{array}\right]=\left[\begin{array}{r}{-4} \\ {2} \\ {4}\end{array}\right]
$$

Key Concepts

Matrix Equations

A matrix equation is an equation in which matrices are used to represent a system of linear relationships among variables. It succinctly expresses multiple linear equations by encapsulating the coefficients in a single matrix multiplied by a variable vector, equated to a constant vector.

System of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. Each equation represents a linear relationship, and solving the system involves finding values for the variables that satisfy every equation simultaneously.

Matrix-Vector Multiplication

Matrix-vector multiplication is the process of taking a matrix and multiplying it by a vector, resulting in a new vector. This operation combines rows of the matrix with the elements of the vector to produce individual equations where each entry in the resulting vector corresponds to a linear combination (dot product) of a row of the matrix and the vector.

Coefficient Matrix

The coefficient matrix is the matrix formed by the coefficients of the variables in a system of linear equations. Each row of the coefficient matrix represents the coefficients of a single equation in the system, and rewriting a matrix equation as a system of equations involves breaking down this matrix into separate scalar equations.

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