Find the determinants. Det $$ \left[\begin{array}{ccc}1 & 0 & -5 \ -5 & 0 & -2 \ -5 & 0 & -1\end{array} ight]= $$ $$ \operatorname{Det}\left[\begin{array}{ccc}1 & 0 & 0 \ -5 & 2 & 0 \ -5 & -2 & -1\end{array} ight]= $$ $$ \operatorname{Det}\left[\begin{array}{ccc}1 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & -1\end{array} ight]= $$ $$ \operatorname{Det}\left[\begin{array}{ccc}1 & -2 & -5 \ 0 & 0 & 0 \ -5 & -2 & -1\end{array} ight] $$

Question

Find the determinants. Det $$ \left[\begin{array}{ccc}1 & 0 & -5 \ -5 & 0 & -2 \ -5 & 0 & -1\end{array}ight]= $$ $$ \operatorname{Det}\left[\begin{array}{ccc}1 & 0 & 0 \ -5 & 2 & 0 \ -5 & -2 & -1\end{array}ight]= $$ $$ \operatorname{Det}\left[\begin{array}{ccc}1 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & -1\end{array}ight]= $$ $$ \operatorname{Det}\left[\begin{array}{ccc}1 & -2 & -5 \ 0 & 0 & 0 \ -5 & -2 & -1\end{array}ight] $$

UpStudy AI · Accepted Answer

1. For 
   $$
   \det\begin{pmatrix} 1 & 0 & -5 \ -5 & 0 & -2 \ -5 & 0 & -1 \end{pmatrix},
   $$
   note that the entire second column is zeros. Hence, the determinant is
   $$
   0.
   $$

2. For 
   $$
   \det\begin{pmatrix} 1 & 0 & 0 \ -5 & 2 & 0 \ -5 & -2 & -1 \end{pmatrix},
   $$
   the matrix is lower triangular (all entries above the main diagonal are zero). The determinant of a triangular matrix is the product of its diagonal entries:
   $$
   1 \cdot 2 \cdot (-1) = -2.
   $$

3. For 
   $$
   \det\begin{pmatrix} 1 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & -1 \end{pmatrix},
   $$
   this diagonal matrix has determinant equal to the product of its diagonal entries:
   $$
   1 \cdot 2 \cdot (-1) = -2.
   $$

4. For 
   $$
   \det\begin{pmatrix} 1 & -2 & -5 \ 0 & 0 & 0 \ -5 & -2 & -1 \end{pmatrix},
   $$
   the second row is entirely zero, so the determinant is
   $$
   0.
   $$

Final answers:
- $$\det\begin{pmatrix} 1 & 0 & -5 \ -5 & 0 & -2 \ -5 & 0 & -1 \end{pmatrix} = 0$$
- $$\det\begin{pmatrix} 1 & 0 & 0 \ -5 & 2 & 0 \ -5 & -2 & -1 \end{pmatrix} = -2$$
- $$\det\begin{pmatrix} 1 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & -1 \end{pmatrix} = -2$$
- $$\det\begin{pmatrix} 1 & -2 & -5 \ 0 & 0 & 0 \ -5 & -2 & -1 \end{pmatrix} = 0$$
The determinants are: 1. 0 2. -2 3. -2 4. 0

Find the determinants.
Det

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