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Expert-verified Found in: Page 289 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10.8. $\left[\begin{array}{ccccc}0& 0& 0& 0& 2\\ 1& 0& 0& 0& 3\\ 0& 1& 0& 0& 4\\ 0& 0& 1& 0& 5\\ 0& 0& 0& 1& 6\end{array}\right]$

Therefore, the determinant of given matrix is given by,

$\mathrm{det}\mathrm{A}=2$

See the step by step solution

## Step 1: Definition

Gaussian elimination method is used to solve a system of linear equations.

Gaussian elimination provides a relatively efficient way of constructing the inverse to a matrix.

Interchanging two rows. Multiplying a row by a constant (any constant which is not zero).

## Step 2: Given

Given Matrix,

${A}_{1}=\left[\begin{array}{ccccc}0& 0& 0& 0& 2\\ 1& 0& 0& 0& 3\\ 0& 1& 0& 0& 4\\ 0& 0& 1& 0& 5\\ 0& 0& 0& 1& 6\end{array}\right]$

## Step 3: To find determinant by using Gaussian Eliminations

We do row interchanges in the following order: first and second row, second and third row, third and fourth row, fourth and fifth row.

Eventually, we get the matrix

${A}_{1}=\left[\begin{array}{ccccc}0& 0& 0& 0& 3\\ 1& 0& 0& 0& 4\\ 0& 0& 1& 0& 5\\ 0& 0& 0& 1& 6\\ 0& 0& 0& 0& 2\end{array}\right]$

We had two row interchanges, so

$\mathrm{det}\mathrm{A}={\left(-1\right)}^{4}.1.1.1.1.2\phantom{\rule{0ex}{0ex}}\mathrm{det}\mathrm{A}=2$ ### Want to see more solutions like these? 