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1. [1 pt each] Compute the following determinants. (a) $\left|\begin{array}{cc...

Feb 3, 2024

1. [1 pt each] Compute the following determinants. (a)

\left|\begin{array}{ccc}2 &amp; 1 &amp; 3 \\ 1 &amp; 0 &amp; 2 \\ 2 &amp; 0 &amp; -2\end{array}\right|

(b)

\left|\begin{array}{cccc}-1 &amp; 3 &amp; -2 &amp; 4 \\ 1 &amp; 6 &amp; 4 &amp; 8 \\ 1 &amp; 3 &amp; 0 &amp; 0 \\ 2 &amp; 6 &amp; 8 &amp; -8\end{array}\right|

(c)

\left|\begin{array}{llll}5 &amp; 2 &amp; 7 &amp; 3 \\ 6 &amp; 2 &amp; 6 &amp; 4 \\ 7 &amp; 4 &amp; 6 &amp; 5 \\ 1 &amp; 6 &amp; 4 &amp; 7\end{array}\right|

(d)

\left|\begin{array}{lllll}3 &amp; 2 &amp; 0 &amp; 0 &amp; 6 \\ 0 &amp; 1 &amp; 0 &amp; 0 &amp; 0 \\ 5 &amp; 2 &amp; 1 &amp; 7 &amp; 1 \\ 0 &amp; 6 &amp; 0 &amp; 0 &amp; 2 \\ 2 &amp; 6 &amp; 3 &amp; 0 &amp; 1\end{array}\right|

(e)

\left|\begin{array}{lllll}1 &amp; 1 &amp; 1 &amp; 1 &amp; 1 \\ 1 &amp; 2 &amp; 2 &amp; 2 &amp; 2 \\ 1 &amp; 1 &amp; 3 &amp; 3 &amp; 3 \\ 1 &amp; 1 &amp; 1 &amp; 4 &amp; 4 \\ 1 &amp; 1 &amp; 2 &amp; 1 &amp; 5\end{array}\right|

2. [1 pt each] Suppose that

a, b, c, d, e

, and

f

are numbers such that

\left|\begin{array}{lll} a &amp; 1 &amp; d \\ b &amp; 1 &amp; e \\ c &amp; 1 &amp; f \end{array}\right|=-7 \text { and }\left|\begin{array}{ccc} a &amp; 1 &amp; d \\ b &amp; 2 &amp; e \\ c &amp; 3 &amp; f \end{array}\right|=5 .

Find (a)

\left|\begin{array}{lll}a &amp; 3 &amp; d \\ b &amp; 3 &amp; e \\ c &amp; 3 &amp; f\end{array}\right|

(b)

\left|\begin{array}{lll}a &amp; 3 &amp; d \\ b &amp; 5 &amp; e \\ c &amp; 7 &amp; f\end{array}\right|

(c)

\left|\begin{array}{lll}e &amp; b &amp; 5 \\ f &amp; c &amp; 9 \\ d &amp; a &amp; 1\end{array}\right|