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Found in: Page 165

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# Compute the determinants of the elementary matrices given in Exercise 25-30.30. \left[ {\begin{aligned}{*{20}{c}}0&1&0\\1&0&0\\0&0&1\end{aligned}} \right].

The determinant of the matrix is $$- 1$$.

See the step by step solution

## Step 1: Compute the determinant of the elementary matrices

Theorem 1 states that the determinant of an $$n \times n$$ matrix A can be computed by cofactor expansion across any row or down any column. Expansion across the

$$i{\mathop{\rm th}\nolimits}$$ row using cofactors in $${C_{ij}} = {\left( { - 1} \right)^{i + j}}\det \,{A_{ij}}$$ gives $$\det \,A = {a_{i1}}{C_{i1}} + {a_{i2}}{C_{i2}} + ... + {a_{in}}{C_{in}}$$.

Cofactor expansion across row 1 is shown below:

\begin{aligned}{c}\left| {\begin{aligned}{*{20}{c}}0&1&0\\1&0&0\\0&0&1\end{aligned}} \right| = - 1\left| {\begin{aligned}{*{20}{c}}1&0\\0&1\end{aligned}} \right|\\ = - 1\left( 1 \right)\\ = - 1\end{aligned}

Thus, the determinant of the matrix is $$- 1$$.