Question

Determine whether the statements that follow are true or false, and justify your answer.

42IfA is a 3x3 matrix of rank 3, then the system$\mathit{A}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{x}}\mathbf{=}\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$ must have infinitely many solutions.

Short answer

The statement, “If A is a 3x4 matrix of rank 3 , then the system$A\stackrel{\rightharpoonup }{x}=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$ must have infinitely many solutions.” is true.

Step-by-step solution

Consider the condition 

Consider a different condition.

A matrix of order 3x4 denotes that, it has three equations and four unknowns.

In general, a matrix with a greater number of unknowns compared to the number of equations will be inconsistent.

Conclusion

The matrix A is a 3x4 matrix that has rank 3. This implies that, the matrix has leading . So, this matrix has one free element. Thus, the system $A\stackrel{\rightharpoonup }{x}=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$must have infinitely many solutions.