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Do these matrices have determinant [tex]$0, 1, 2$[/tex], or [tex]$3$[/tex]?

[tex]\[
A=\left[\begin{array}{ccc}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0
\end{array}\right]
\quad
B=\left[\begin{array}{ccc}
0 & 1 & 1 \\
1 & 0 & 1 \\
1 & 1 & 0
\end{array}\right]
\quad
C=\left[\begin{array}{ccc}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{array}\right]
\][/tex]

Sagot :

GinnyAnsweridn
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.
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