Chapter 11.3, Problem 4AYU

To determine

To explain: Whether the statement “When using Cramer’s Rule, if $D=0$ , then the system of linear equations is inconsistent.” is true or false.

Answer to Problem 4AYU

The statement “When using Cramer’s Rule, if $D=0$ , then the system of linear equations is inconsistent.” is false.

Explanation of Solution

Given information:

The statement “When using Cramer’s Rule, if $D=0$ , then the system of linear equations is inconsistent.”

Consider the provided statement “When using Cramer’s Rule, if $D=0$ , then the system of linear equations is inconsistent.”

A system of equations is said to be inconsistent and solution set is $\varphi \text{ or }\left\{\right\}$ when $D=0$ and at least one of the determinants $D_x,D_y\text{ or }{D}_z$ is different from zero.

A system of equations is said to be consistent and dependent with infinitely many solution when $D=0$ and all determinants $D_x,D_y\text{ and }{D}_z$ are zero.

Therefore, there are two possible cases when $D=0$ to solve system of system with help of Cramer’s Rule.

Thus, the statement “When using Cramer’s Rule, if $D=0$ , then the system of linear equations is inconsistent.” is false.