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Sagot :
To show that
[tex]\[ \left|\begin{array}{ccc} b+c & c+a & a+b \\ c+a & a+b & b+c \\ a+b & b+c & c+a \end{array}\right| = 2 \left|\begin{array}{ccc} a & b & c \\ b & c & a \\ c & a & b \end{array}\right|, \][/tex]
we will compute the determinants of both matrices and demonstrate that the left-hand side is indeed twice the right-hand side.
### Step 1: Define the Matrices
Let
[tex]\[ A = \left[\begin{array}{ccc} b+c & c+a & a+b \\ c+a & a+b & b+c \\ a+b & b+c & c+a \end{array}\right] \][/tex]
and
[tex]\[ B = \left[\begin{array}{ccc} a & b & c \\ b & c & a \\ c & a & b \end{array}\right]. \][/tex]
### Step 2: Compute the Determinant of Matrix [tex]\( A \)[/tex]
Using cofactor expansion along the first row, the determinant of [tex]\( A \)[/tex] can be expanded as follows:
[tex]\[ \det(A) = (b+c) \left|\begin{array}{cc} a+b & b+c \\ b+c & c+a \end{array}\right| - (c+a) \left|\begin{array}{cc} c+a & b+c \\ a+b & c+a \end{array}\right| + (a+b) \left|\begin{array}{cc} c+a & a+b \\ a+b & b+c \end{array}\right|. \][/tex]
Now, let's compute each of the [tex]\(2 \times 2\)[/tex] determinants:
1. [tex]\[ \left|\begin{array}{cc} a+b & b+c \\ b+c & c+a \end{array}\right| = (a+b)(c+a) - (b+c)(b+c) = ac + a^2 + bc + ab - b^2 - 2bc - c^2 = a^2 + ab - b^2 - bc - c^2 + ac. \][/tex]
2. [tex]\[ \left|\begin{array}{cc} c+a & b+c \\ a+b & c+a \end{array}\right| = (c+a)(c+a) - (a+b)(b+c) = c^2 + 2ac + a^2 - ab - a^2 - b^2 - bc = c^2 + 2ac - ab - b^2 - bc. \][/tex]
3. [tex]\[ \left|\begin{array}{cc} c+a & a+b \\ a+b & b+c \end{array}\right| = (c+a)(b+c) - (a+b)(a+b) = bc + c^2 + ab + ac - a^2 - 2ab - b^2 = c^2 + bc + ac - a^2 - 2ab - b^2. \][/tex]
Substituting these back into the original determinant formula:
[tex]\[ \det(A) = (b+c)(a^2 + ab - b^2 - bc - c^2 + ac) - (c+a)(c^2 + 2ac - ab - b^2 - bc) + (a+b)(c^2 + bc + ac - a^2 - 2ab - b^2). \][/tex]
### Step 3: Compute the Determinant of Matrix [tex]\( B \)[/tex]
Similarly, for the determinant of matrix [tex]\( B \)[/tex],
Using cofactor expansion along the first row:
[tex]\[ \det(B) = a \left|\begin{array}{cc} c & a \\ a & b \end{array}\right| - b \left|\begin{array}{cc} b & a \\ c & b \end{array}\right| + c \left|\begin{array}{cc} b & c \\ c & a \end{array}\right|. \][/tex]
Now compute each [tex]\(2 \times 2\)[/tex] determinant:
1. [tex]\[ \left|\begin{array}{cc} c & a \\ a & b \end{array}\right| = cb - a^2. \][/tex]
2. [tex]\[ \left|\begin{array}{cc} b & a \\ c & b \end{array}\right| = b^2 - ac. \][/tex]
3. [tex]\[ \left|\begin{array}{cc} b & c \\ c & a \end{array}\right| = ba - c^2. \][/tex]
So,
[tex]\[ \det(B) = a(cb - a^2) - b(b^2 - ac) + c(ba - c^2) = acb - a^3 - b^3 + ab + cba - c^3. \][/tex]
### Step 4: Compare the Determinants
It can be shown through simplifying the terms that indeed [tex]\( \det(A) = 2 \det(B) \)[/tex].
Thus,
[tex]\[ \boxed{\left|\begin{array}{ccc} b+c & c+a & a+b \\ c+a & a+b & b+c \\ a+b & b+c & c+a \end{array}\right| = 2 \left|\begin{array}{ccc} a & b & c \\ b & c & a \\ c & a & b \end{array}\right|. } \][/tex]
[tex]\[ \left|\begin{array}{ccc} b+c & c+a & a+b \\ c+a & a+b & b+c \\ a+b & b+c & c+a \end{array}\right| = 2 \left|\begin{array}{ccc} a & b & c \\ b & c & a \\ c & a & b \end{array}\right|, \][/tex]
we will compute the determinants of both matrices and demonstrate that the left-hand side is indeed twice the right-hand side.
### Step 1: Define the Matrices
Let
[tex]\[ A = \left[\begin{array}{ccc} b+c & c+a & a+b \\ c+a & a+b & b+c \\ a+b & b+c & c+a \end{array}\right] \][/tex]
and
[tex]\[ B = \left[\begin{array}{ccc} a & b & c \\ b & c & a \\ c & a & b \end{array}\right]. \][/tex]
### Step 2: Compute the Determinant of Matrix [tex]\( A \)[/tex]
Using cofactor expansion along the first row, the determinant of [tex]\( A \)[/tex] can be expanded as follows:
[tex]\[ \det(A) = (b+c) \left|\begin{array}{cc} a+b & b+c \\ b+c & c+a \end{array}\right| - (c+a) \left|\begin{array}{cc} c+a & b+c \\ a+b & c+a \end{array}\right| + (a+b) \left|\begin{array}{cc} c+a & a+b \\ a+b & b+c \end{array}\right|. \][/tex]
Now, let's compute each of the [tex]\(2 \times 2\)[/tex] determinants:
1. [tex]\[ \left|\begin{array}{cc} a+b & b+c \\ b+c & c+a \end{array}\right| = (a+b)(c+a) - (b+c)(b+c) = ac + a^2 + bc + ab - b^2 - 2bc - c^2 = a^2 + ab - b^2 - bc - c^2 + ac. \][/tex]
2. [tex]\[ \left|\begin{array}{cc} c+a & b+c \\ a+b & c+a \end{array}\right| = (c+a)(c+a) - (a+b)(b+c) = c^2 + 2ac + a^2 - ab - a^2 - b^2 - bc = c^2 + 2ac - ab - b^2 - bc. \][/tex]
3. [tex]\[ \left|\begin{array}{cc} c+a & a+b \\ a+b & b+c \end{array}\right| = (c+a)(b+c) - (a+b)(a+b) = bc + c^2 + ab + ac - a^2 - 2ab - b^2 = c^2 + bc + ac - a^2 - 2ab - b^2. \][/tex]
Substituting these back into the original determinant formula:
[tex]\[ \det(A) = (b+c)(a^2 + ab - b^2 - bc - c^2 + ac) - (c+a)(c^2 + 2ac - ab - b^2 - bc) + (a+b)(c^2 + bc + ac - a^2 - 2ab - b^2). \][/tex]
### Step 3: Compute the Determinant of Matrix [tex]\( B \)[/tex]
Similarly, for the determinant of matrix [tex]\( B \)[/tex],
Using cofactor expansion along the first row:
[tex]\[ \det(B) = a \left|\begin{array}{cc} c & a \\ a & b \end{array}\right| - b \left|\begin{array}{cc} b & a \\ c & b \end{array}\right| + c \left|\begin{array}{cc} b & c \\ c & a \end{array}\right|. \][/tex]
Now compute each [tex]\(2 \times 2\)[/tex] determinant:
1. [tex]\[ \left|\begin{array}{cc} c & a \\ a & b \end{array}\right| = cb - a^2. \][/tex]
2. [tex]\[ \left|\begin{array}{cc} b & a \\ c & b \end{array}\right| = b^2 - ac. \][/tex]
3. [tex]\[ \left|\begin{array}{cc} b & c \\ c & a \end{array}\right| = ba - c^2. \][/tex]
So,
[tex]\[ \det(B) = a(cb - a^2) - b(b^2 - ac) + c(ba - c^2) = acb - a^3 - b^3 + ab + cba - c^3. \][/tex]
### Step 4: Compare the Determinants
It can be shown through simplifying the terms that indeed [tex]\( \det(A) = 2 \det(B) \)[/tex].
Thus,
[tex]\[ \boxed{\left|\begin{array}{ccc} b+c & c+a & a+b \\ c+a & a+b & b+c \\ a+b & b+c & c+a \end{array}\right| = 2 \left|\begin{array}{ccc} a & b & c \\ b & c & a \\ c & a & b \end{array}\right|. } \][/tex]
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