Question 6.2.4: Find det[0 7 5 3 1 1 2 1 1 1 2 −1 1 1 1 2]....

Linear Algebra with Applications [1016048]

Find

$det\begin{bmatrix}0&7&5&3\\1&1&2&1\\1&1&2&−1\\1&1&1&2\end{bmatrix}$ .

Step-by-Step

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We go through the elimination process, keeping a note of all the row swaps and row divisions we perform (if any). In view of part b of Algorithm 6.2.5, we realize that it suffices to reduce A to an upper triangular matrix: There is no need to eliminate entries above the diagonal, or to make the diagonal entries equal to 1.

We have performed two row swaps, so that $\det A = (−1)^2(\det B) = 7(−1)(−2) =14$ . We have used Theorem 6.1.4: The determinant of the triangular matrix B is the product of its diagonal entries.

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