Write each matrix equation as a system of linear equations...

Write each matrix equation as a system of linear equations without matrices.
$$\left[\begin{array}{rrr}2 & 0 & -1 \\0 & 3 & 0 \\1 & 1 & 0 \end{array}\right]\left[\begin{array}{l}x \\y \\z\end{array}\right]=\left[\begin{array}{l}
6 \\9 \\5\end{array}\right]$$

Key Concepts

Coefficient Matrix

The coefficient matrix is the array in a matrix equation that contains the coefficients of the variables found in each linear equation. It organizes the numerical multipliers of each variable across all equations, allowing for a compact representation of the system and facilitating operations such as matrix multiplication and inversion.

Translation from Matrix Equation to System of Equations

Translating a matrix equation into a system of equations involves writing an individual linear equation for each row of the matrix product. This process effectively 'unpacks' the compact matrix form into explicit equations that show how the coefficients combine with the variables to yield the constants on the right-hand side, thus making the structure of the problem easier to understand and solve.

System of Linear Equations

A system of linear equations is a collection of equations in which each equation is linear in the variables. Converting a matrix equation into a system of linear equations involves expressing each row of the product as a separate linear equation, providing a clear path for algebraic methods like substitution or elimination to find the solution.

Matrix Multiplication

Matrix multiplication is a method of combining two arrays, such as a coefficient matrix and a variable vector, to produce a new vector. In the context of systems of equations, each element of the resulting vector is obtained by taking the dot product of a row from the coefficient matrix with the column vector of variables, which then equates to the corresponding constant from the right-hand side of the equation.

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