Question

Write the augmented matrix for each system and give its dimension. Do not solve. 2x + y + z - 3 = 0 3x - 4y + 2z + 7 = 0 x + y + z - 2 = 0

Accepted Answer

Rewrite each equation in the system so that all variables and constants are on the same side, isolating the constant term on the right side. For example, rewrite the first equation as $$2x + y + z = 3. $$Identify the coefficients of each variable in each equation and the constants on the right side. For the system, the coefficients form the matrix part, and the constants form the augmented column. Construct the augmented matrix by placing the coefficients of $$x$$, $$y$$, and $$z in $$rows corresponding to each equation, and append the constants as the last column. The matrix will look like this:  
$$\left[\begin{array}{ccc|c} 2 & 1 & 1 & 3 \ 3 & -4 & 2 & -7 \ 1 & 1 & 1 & 2 \end{array}ight]$$ Determine the dimension of the augmented matrix by counting the number of rows and columns. The number of rows corresponds to the number of equations, and the number of columns corresponds to the number of variables plus one (for the augmented part). Summarize the dimension as $$m 	imes n$$, where $$m is $$the number of equations (rows) and $$n is $$the number of variables plus one (columns). In this case, it will be $$3 	imes 4.$$

Write the augmented matrix for each system and give its dimension. Do not solve.
2x + y + z - 3 = 0
3x - 4y + 2z + 7 = 0
x + y + z - 2 = 0

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주요 개념

Augmented Matrix

Matrix Dimensions

System of Linear Equations