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Found in: Page 39

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Determine whether the statements that follow are true or false, and justify your answer.15: The system ${\mathbf{A}}\stackrel{\to }{\mathbf{x}}{\mathbf{=}}\mathbf{\left[}\begin{array}{c}\mathbf{0}\\ \mathbf{0}\\ \mathbf{0}\\ \mathbf{1}\end{array}\mathbf{\right]}$inconsistent for all ${\mathbf{4}}{\mathbf{×}}{\mathbf{3}}$matrices A.

True, the given system of equations is inconsistent.

See the 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.

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

## Step 2: Justification of answer

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

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

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 $\stackrel{\to }{x}$the value$0\ne 1$

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