Question

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

15: The system$\mathbf{A}\overrightarrow{\mathbf{x}}\mathbf{=}\mathbf{\left[}\begin{array}{c}\mathbf{0}\\ \mathbf{0}\\ \mathbf{0}\\ \mathbf{1}\end{array}\mathbf{\right]}$inconsistent for all $\mathbf{4}\mathbf{\times }\mathbf{3}$matrices A.

Short answer

True, the given system of equations is inconsistent.

Step-by-step solution

Step 1:Corresponding augmented matrix

We have given that,a system of equationslets the A matrix be

$\left[\begin{array}{ccc}a& e& i\\ b& f& j\\ c& g& k\\ d& h& l\end{array}\right]$

Now, we have given the system.

$\mathrm{A}\overrightarrow{\mathrm{x}}=\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right]$

Justification of answer

Put the value of A matrix in the above equation we will get.

$\left[\begin{array}{ccc}a& e& i\\ b& f& j\\ c& g& k\\ d& h& l\end{array}\right]\overrightarrow{\mathrm{x}}=\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right]$

Now since the matrix A has 3 columns then it has a maximum of 3 pivot entry.

Also, the pivot entry not possible in the last row.

This means for any value of $\overrightarrow{x}$the value$0\ne 1$

Hence, the given statement is true, because the given system is inconsistent.