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Sagot :
To solve the given determinant equation for [tex]\( x \)[/tex]:
[tex]\[ \left|\begin{array}{ccc} 7 & 2 & -1 \\ 3x & 2 & 2 \\ -3 & 1 & -7 \end{array}\right| = 65 \][/tex]
we need to follow these steps:
1. Calculate the Determinant of the Matrix:
We start by calculating the determinant of the [tex]\( 3 \times 3 \)[/tex] matrix:
[tex]\[ A = \begin{vmatrix} 7 & 2 & -1 \\ 3x & 2 & 2 \\ -3 & 1 & -7 \end{vmatrix} \][/tex]
Using the rule for finding the determinant of a [tex]\( 3 \times 3 \)[/tex] matrix, we have:
[tex]\[ \text{det}(A) = 7 \begin{vmatrix} 2 & 2 \\ 1 & -7 \end{vmatrix} - 2 \begin{vmatrix} 3x & 2 \\ -3 & -7 \end{vmatrix} - 1 \begin{vmatrix} 3x & 2 \\ -3 & 1 \end{vmatrix} \][/tex]
Now, calculate the [tex]\( 2 \times 2 \)[/tex] determinants:
[tex]\[ \begin{vmatrix} 2 & 2 \\ 1 & -7 \end{vmatrix} = (2)(-7) - (2)(1) = -14 - 2 = -16 \][/tex]
[tex]\[ \begin{vmatrix} 3x & 2 \\ -3 & -7 \end{vmatrix} = (3x)(-7) - (2)(-3) = -21x + 6 \][/tex]
[tex]\[ \begin{vmatrix} 3x & 2 \\ -3 & 1 \end{vmatrix} = (3x)(1) - (2)(-3) = 3x + 6 \][/tex]
Substitute these back into the determinant formula:
[tex]\[ \text{det}(A) = 7(-16) - 2(-21x + 6) - 1(3x + 6) \][/tex]
Simplify:
[tex]\[ \text{det}(A) = -112 + 42x - 12 - 3x - 6 \][/tex]
Combine like terms:
[tex]\[ \text{det}(A) = 39x - 130 \][/tex]
2. Set the Determinant Equal to 65:
We are given that the determinant is equal to 65. So, we set up the equation:
[tex]\[ 39x - 130 = 65 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], add 130 to both sides of the equation:
[tex]\[ 39x = 195 \][/tex]
Now, divide by 39:
[tex]\[ x = \frac{195}{39} = 5 \][/tex]
Thus, the solution to the equation is:
[tex]\[ x = 5 \][/tex]
So, we have calculated that the determinant of the matrix equals [tex]\( 39x - 130 \)[/tex], and solving for [tex]\( x \)[/tex] when this determinant is set to 65, we find:
[tex]\[ x = 5 \][/tex]
[tex]\[ \left|\begin{array}{ccc} 7 & 2 & -1 \\ 3x & 2 & 2 \\ -3 & 1 & -7 \end{array}\right| = 65 \][/tex]
we need to follow these steps:
1. Calculate the Determinant of the Matrix:
We start by calculating the determinant of the [tex]\( 3 \times 3 \)[/tex] matrix:
[tex]\[ A = \begin{vmatrix} 7 & 2 & -1 \\ 3x & 2 & 2 \\ -3 & 1 & -7 \end{vmatrix} \][/tex]
Using the rule for finding the determinant of a [tex]\( 3 \times 3 \)[/tex] matrix, we have:
[tex]\[ \text{det}(A) = 7 \begin{vmatrix} 2 & 2 \\ 1 & -7 \end{vmatrix} - 2 \begin{vmatrix} 3x & 2 \\ -3 & -7 \end{vmatrix} - 1 \begin{vmatrix} 3x & 2 \\ -3 & 1 \end{vmatrix} \][/tex]
Now, calculate the [tex]\( 2 \times 2 \)[/tex] determinants:
[tex]\[ \begin{vmatrix} 2 & 2 \\ 1 & -7 \end{vmatrix} = (2)(-7) - (2)(1) = -14 - 2 = -16 \][/tex]
[tex]\[ \begin{vmatrix} 3x & 2 \\ -3 & -7 \end{vmatrix} = (3x)(-7) - (2)(-3) = -21x + 6 \][/tex]
[tex]\[ \begin{vmatrix} 3x & 2 \\ -3 & 1 \end{vmatrix} = (3x)(1) - (2)(-3) = 3x + 6 \][/tex]
Substitute these back into the determinant formula:
[tex]\[ \text{det}(A) = 7(-16) - 2(-21x + 6) - 1(3x + 6) \][/tex]
Simplify:
[tex]\[ \text{det}(A) = -112 + 42x - 12 - 3x - 6 \][/tex]
Combine like terms:
[tex]\[ \text{det}(A) = 39x - 130 \][/tex]
2. Set the Determinant Equal to 65:
We are given that the determinant is equal to 65. So, we set up the equation:
[tex]\[ 39x - 130 = 65 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], add 130 to both sides of the equation:
[tex]\[ 39x = 195 \][/tex]
Now, divide by 39:
[tex]\[ x = \frac{195}{39} = 5 \][/tex]
Thus, the solution to the equation is:
[tex]\[ x = 5 \][/tex]
So, we have calculated that the determinant of the matrix equals [tex]\( 39x - 130 \)[/tex], and solving for [tex]\( x \)[/tex] when this determinant is set to 65, we find:
[tex]\[ x = 5 \][/tex]
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