The augmented matrix of a linear system with three variables and four equations has rank of 4. What can you say about the solutions of this system?

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Say the system is

`a_{11}x+a_{12}y+a_{13}z=b_1`

`a_{21}x+a_{22}y+a_{23}z=b_2`

`a_{31}x+a_{32}y+a_{33}z=b_3`

`a_{41}x+a_{42}y+a_{43}z=b_4,`

so the augmented matrix is

`[[a_{11},a_{12},a_{13},b_1],[a_{21},a_{22},a_{23},b_2],[a_{31},a_{32},a_{33},b_3],[a_{41},a_{42},a_{43},b_4]].`

For this to have rank 4, the columns must be independent, so no linear combination of the first three columns can result in the fourth column. In other words, there is no solution to the system.

...

Unlock
This Answer Now

Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.

Start your 48-Hour Free Trial

Say the system is

`a_{11}x+a_{12}y+a_{13}z=b_1`

`a_{21}x+a_{22}y+a_{23}z=b_2`

`a_{31}x+a_{32}y+a_{33}z=b_3`

`a_{41}x+a_{42}y+a_{43}z=b_4,`

so the augmented matrix is

`[[a_{11},a_{12},a_{13},b_1],[a_{21},a_{22},a_{23},b_2],[a_{31},a_{32},a_{33},b_3],[a_{41},a_{42},a_{43},b_4]].`

For this to have rank 4, the columns must be independent, so no linear combination of the first three columns can result in the fourth column. In other words, there is no solution to the system.

This is a special case of the theorem described in the link, which says that a system of equations has a solution if and only if the rank of the coefficient matrix equals the rank of the augmented matrix. Here, the largest rank the coefficient matrix can have is 3 (since there are 3 columns), so there can be no solution.

 

Approved by eNotes Editorial Team