We are given a classmate's matrix representation to a system. Our classmate says that x is equal to 2 and y is equal to 0. We will explain and correct the error in the representation.
To do so, let's first recall how to represent a system of equations with a matrix. In a matrix, each row represents an equation written in standard form. Each column other than the last one demonstrates the coefficients of one of the variables. The last column shows the constants of the equations.
ax+ by= c dx+ ey= f
⇒
[
cc|c
a & b & c
d & e & f
]
Note that we draw a vertical bar between the coefficients and constants to replace the equal signs. Now we can write the solutions of the given system as equations in standard form.
Solutions
x= 2
y= 0
Equation Forms
1x+ 0y= 2
0x+ 1y= 0
Next we will write these equations as a system and transform them into a matrix.
1x+ 0y= 2 0x+ 1y= 0
⇒
[
cc|c
1 & 0 & 2
0 & 1 & 0
]
Now we will compare the matrices. Our classmate forgot to draw the vertical bar that represents the equals sign, so we will draw it. In addition, in the second row of the given matrix the coefficients of both x and y are 0.
This means that we have an identity, 0=0, so there are infinitely many solutions for the system. To obtain y=0 the element in the second column and the second row needs to be 1. Finally, we will correct the given solution according to our findings.
