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 given: $\left[\begin{array}{ccc}2& 0& -1\\ 0& 3& 0\\ 1& 1& 0\end{array}\right]\cdot [x,y,z]=[6,9,5]$ represents a system of linear equations where the matrix multiplies the vector of variables $x,y,z$ to produce the result vector.
2. Write the multiplication of the first row of the matrix by the variable vector: $2x\; +\; 0y\; +\; (-1)z$. This equals the first element of the result vector, which is 6. So, the first equation is $2x\; -\; z\; =\; 6$.
3. Write the multiplication of the second row of the matrix by the variable vector: $0x\; +\; 3y\; +\; 0z$. This equals the second element of the result vector, which is 9. So, the second equation is $3y\; =\; 9$.
4. Write the multiplication of the third row of the matrix by the variable vector: $1x\; +\; 1y\; +\; 0z$. This equals the third element of the result vector, which is 5. So, the third equation is $x\; +\; y\; =\; 5$.
5. Combine all three equations to form the system of linear equations without matrices: $\backslash begin\{cases\}\; 2x\; -\; z\; =\; 6\; \backslash \backslash \; 3y\; =\; 9\; \backslash \backslash \; x\; +\; y\; =\; 5\; \backslash end\{cases\}$.

Key Concepts

Matrix Multiplication

Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix and summing the products. In this context, multiplying the coefficient matrix by the variable matrix results in a new matrix representing the system's outputs.

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same variables. Each row in the matrix equation corresponds to one linear equation, which can be written by equating the sum of products of coefficients and variables to the constants.

Matrix Equation to Linear System Conversion

Converting a matrix equation to a system of linear equations means expressing the matrix multiplication as individual equations. Each element in the resulting matrix equals the sum of products from the corresponding row and variable vector, forming separate linear equations.