Question

Consider the accompanying table. For some linear systems$\mathbf{A}\mathbf{=}\overrightarrow{\mathbf{x}}\mathbf{=}\overrightarrow{\mathbf{b}}$, you are given either the rank of the coefficient matrix$\mathbf{A}$ , or the rank of the augmented matrix $[A:\overrightarrow{b}]$. In each case, state whether the system could have no solution, one solution, or infinitely many solutions. There may be more than one possibility for some systems. Justify your answers.

Short answer

a. Infinitely many solutions or no solution for the system with three equations and four unknowns.

b. One solution or no solution for the system with four equations and three unknowns.

c. No solution for the system with four equations and three unknowns.

d. Infinitely many solutions for the system with three equations and four unknowns.

Step-by-step solution

Consider the system.

Coefficient matrix consists of coefficients of the system of linear equations.

Augmented matrix consists of coefficient matrix along with a column consisting of the values of the linear equations.

Consider a linear system,

$A\overrightarrow{x}=\overrightarrow{b}$

Determine the solution for the system.

Suppose that an $m\times n$system of linear equations is given. That is, there are $m$ linear equations and $n$unknowns and the rank $r$.

If $m<n$, then the system has no solution or it has infinitely many solutions.

Suppose the system is consistent. Then the rank $r$ of the system satisfies $r\le n$. Also, the system has $n-r$free variables.

Suppose the system is consistent. If $n=r$, then the system has a unique solution. If $n>r$, then the system has infinitely many solutions.

a. As the number of equations is less than the number of unknowns, so, the system with three equations and four unknowns will have infinitely many solutions or no solution.

b. As the number of unknowns equals the rank of the matrix, thus, the system will have one solution. As there are no free variables, thus, the system will have no solution.

So, the system with four equations and three unknowns and rank being three, it will have one solution or no solution.

c. As the number of equations is greater than the number of unknowns and the rank of$\left[A:\overrightarrow{b}\right]$ is greater than the number of unknowns, so, the system with four equations and three unknowns will have no solution.

d. As the number of unknowns is greater than the rank of the matrix, so, the system with three equations and four unknowns will have infinitely many solutions.

Final answer

a. For the system with three equations and four unknowns, infinitely many solutions or no solution.

b. For the system with four equations and three unknowns, one solution or no solution.

c. For the system with four equations and three unknowns, no solution.

d. For the system with three equations and four unknowns, infinitely many solutions.