In Exercises 33 - 36, write each matrix equation as a system of linear equations without matrices.

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1. Identify the matrix equation: \( \begin{bmatrix} 2 & 0 & -1 \\ 0 & 3 & 0 \\ 1 & 1 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 9 \\ 5 \end{bmatrix} \).
2. Multiply the first row of the matrix by the column vector: \( 2x + 0y - 1z = 6 \).
3. Multiply the second row of the matrix by the column vector: \( 0x + 3y + 0z = 9 \).
4. Multiply the third row of the matrix by the column vector: \( 1x + 1y + 0z = 5 \).
5. Write the system of linear equations: \( 2x - z = 6 \), \( 3y = 9 \), \( x + y = 5 \).

Key Concepts

Matrix Representation

A matrix is a rectangular array of numbers arranged in rows and columns. In the context of linear equations, matrices can represent coefficients of variables in a system. The matrix equation Ax = b combines the coefficient matrix A, the variable matrix x, and the constant matrix b, allowing for a compact representation of multiple linear equations.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations that share the same variables. The goal is to find values for these variables that satisfy all equations simultaneously. Each equation can be represented in the form ax + by + cz = d, where a, b, c, and d are constants, and x, y, z are the variables.

Conversion from Matrix to Linear Equations

To convert a matrix equation into a system of linear equations, each row of the coefficient matrix corresponds to a separate equation. The elements of each row represent the coefficients of the variables, while the elements of the result matrix represent the constants on the right side of the equations. This process allows for a clear understanding of the relationships between the variables.