How to explain the why of augmented matrix is consistent without performing any row operation ?

Question

Consider the two systems represented by the augmented matrices.

If the first system is known to be consistent, explain why the second system is also consistent. Do not perform any row operations.

Answer 1

Let A = coefficient matrix for system of linear equations

Let x = column vector of vars

Let B = Column vector of constants

NOTE these two augmented matrices are represented below

1) [A|B1]

2) [A|B2]

case i)

[A|B1] has a unique solution then [A|B2] does not have free variables

[A|B1] has a unique solution then [A|B2] will also have solution

case ii)

[A|B1] has a free variables thus infinity many solutions then [A|B2] has no solutions if the elements in the last row ( expect in last column) augmented matrix [A|B2] has to be zero

In summary your question falls under case i)

Also note you can take the determinant to show LI which denotes unique solution

-Maksim