Let's determine the determinants of the given matrices [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex].
Given matrices:
[tex]\[ A = \begin{pmatrix}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0
\end{pmatrix} \][/tex]
[tex]\[ B = \begin{pmatrix}
0 & 1 & 1 \\
1 & 0 & 1 \\
1 & 1 & 0
\end{pmatrix} \][/tex]
[tex]\[ C = \begin{pmatrix}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{pmatrix} \][/tex]
1. Determinant of Matrix [tex]\(A\)[/tex]:
[tex]\[ A = \begin{pmatrix}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0
\end{pmatrix} \][/tex]
After calculating the determinant, we find:
[tex]\[ \det(A) = 1 \][/tex]
2. Determinant of Matrix [tex]\(B\)[/tex]:
[tex]\[ B = \begin{pmatrix}
0 & 1 & 1 \\
1 & 0 & 1 \\
1 & 1 & 0
\end{pmatrix} \][/tex]
After calculating the determinant, we find:
[tex]\[ \det(B) = 2 \][/tex]
3. Determinant of Matrix [tex]\(C\)[/tex]:
[tex]\[ C = \begin{pmatrix}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{pmatrix} \][/tex]
After calculating the determinant, we find:
[tex]\[ \det(C) = 0 \][/tex]
Thus, the determinants of the matrices [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] are [tex]\( 1 \)[/tex], [tex]\( 2 \)[/tex], and [tex]\( 0 \)[/tex] respectively.
