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Sagot :
Let's solve for the determinants of the given linear system systematically.
We are given the linear system:
[tex]\[ \begin{array}{l} 5x + 2y = 14 \\ -3x - 5y = 3 \end{array} \][/tex]
Step 1: Write the system in matrix form:
This system can be represented in the matrix form [tex]\(A \mathbf{x} = \mathbf{b}\)[/tex], where [tex]\(A\)[/tex] is the coefficient matrix, [tex]\(\mathbf{x}\)[/tex] is the vector of variables, and [tex]\(\mathbf{b}\)[/tex] is the constants vector.
[tex]\[ A = \begin{pmatrix} 5 & 2 \\ -3 & -5 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \\ y \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 14 \\ 3 \end{pmatrix} \][/tex]
Step 2: Find the determinant of matrix [tex]\(A\)[/tex]:
[tex]\[ |A| = \begin{vmatrix} 5 & 2 \\ -3 & -5 \end{vmatrix} \][/tex]
Using the formula for the determinant of a 2x2 matrix, [tex]\(|A| = ad - bc\)[/tex]:
[tex]\[ |A| = (5 \cdot -5) - (2 \cdot -3) = -25 + 6 = -19 \][/tex]
So, the determinant of [tex]\(A\)[/tex] is:
[tex]\[ |A| = -19 \][/tex]
Step 3: Find the determinant of matrix [tex]\(A_x\)[/tex]:
Matrix [tex]\(A_x\)[/tex] is obtained by replacing the first column of [tex]\(A\)[/tex] with the constant vector [tex]\(\mathbf{b}\)[/tex]:
[tex]\[ A_x = \begin{pmatrix} 14 & 2 \\ 3 & -5 \end{pmatrix} \][/tex]
Now, we compute the determinant of [tex]\(A_x\)[/tex]:
[tex]\[ |A_x| = \begin{vmatrix} 14 & 2 \\ 3 & -5 \end{vmatrix} \][/tex]
[tex]\[ |A_x| = (14 \cdot -5) - (2 \cdot 3) = -70 - 6 = -76 \][/tex]
So, the determinant of [tex]\(A_x\)[/tex] is:
[tex]\[ |A_x| = -76 \][/tex]
Step 4: Find the determinant of matrix [tex]\(A_y\)[/tex]:
Matrix [tex]\(A_y\)[/tex] is obtained by replacing the second column of [tex]\(A\)[/tex] with the constant vector [tex]\(\mathbf{b}\)[/tex]:
[tex]\[ A_y = \begin{pmatrix} 5 & 14 \\ -3 & 3 \end{pmatrix} \][/tex]
Now, we compute the determinant of [tex]\(A_y\)[/tex]:
[tex]\[ |A_y| = \begin{vmatrix} 5 & 14 \\ -3 & 3 \end{vmatrix} \][/tex]
[tex]\[ |A_y| = (5 \cdot 3) - (14 \cdot -3) = 15 + 42 = 57 \][/tex]
So, the determinant of [tex]\(A_y\)[/tex] is:
[tex]\[ |A_y| = 57 \][/tex]
In summary, the determinants for the given linear system are:
[tex]\[ \begin{array}{l} |A| = -19 \\ |A_x| = -76 \\ |A_y| = 57 \end{array} \][/tex]
We are given the linear system:
[tex]\[ \begin{array}{l} 5x + 2y = 14 \\ -3x - 5y = 3 \end{array} \][/tex]
Step 1: Write the system in matrix form:
This system can be represented in the matrix form [tex]\(A \mathbf{x} = \mathbf{b}\)[/tex], where [tex]\(A\)[/tex] is the coefficient matrix, [tex]\(\mathbf{x}\)[/tex] is the vector of variables, and [tex]\(\mathbf{b}\)[/tex] is the constants vector.
[tex]\[ A = \begin{pmatrix} 5 & 2 \\ -3 & -5 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \\ y \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 14 \\ 3 \end{pmatrix} \][/tex]
Step 2: Find the determinant of matrix [tex]\(A\)[/tex]:
[tex]\[ |A| = \begin{vmatrix} 5 & 2 \\ -3 & -5 \end{vmatrix} \][/tex]
Using the formula for the determinant of a 2x2 matrix, [tex]\(|A| = ad - bc\)[/tex]:
[tex]\[ |A| = (5 \cdot -5) - (2 \cdot -3) = -25 + 6 = -19 \][/tex]
So, the determinant of [tex]\(A\)[/tex] is:
[tex]\[ |A| = -19 \][/tex]
Step 3: Find the determinant of matrix [tex]\(A_x\)[/tex]:
Matrix [tex]\(A_x\)[/tex] is obtained by replacing the first column of [tex]\(A\)[/tex] with the constant vector [tex]\(\mathbf{b}\)[/tex]:
[tex]\[ A_x = \begin{pmatrix} 14 & 2 \\ 3 & -5 \end{pmatrix} \][/tex]
Now, we compute the determinant of [tex]\(A_x\)[/tex]:
[tex]\[ |A_x| = \begin{vmatrix} 14 & 2 \\ 3 & -5 \end{vmatrix} \][/tex]
[tex]\[ |A_x| = (14 \cdot -5) - (2 \cdot 3) = -70 - 6 = -76 \][/tex]
So, the determinant of [tex]\(A_x\)[/tex] is:
[tex]\[ |A_x| = -76 \][/tex]
Step 4: Find the determinant of matrix [tex]\(A_y\)[/tex]:
Matrix [tex]\(A_y\)[/tex] is obtained by replacing the second column of [tex]\(A\)[/tex] with the constant vector [tex]\(\mathbf{b}\)[/tex]:
[tex]\[ A_y = \begin{pmatrix} 5 & 14 \\ -3 & 3 \end{pmatrix} \][/tex]
Now, we compute the determinant of [tex]\(A_y\)[/tex]:
[tex]\[ |A_y| = \begin{vmatrix} 5 & 14 \\ -3 & 3 \end{vmatrix} \][/tex]
[tex]\[ |A_y| = (5 \cdot 3) - (14 \cdot -3) = 15 + 42 = 57 \][/tex]
So, the determinant of [tex]\(A_y\)[/tex] is:
[tex]\[ |A_y| = 57 \][/tex]
In summary, the determinants for the given linear system are:
[tex]\[ \begin{array}{l} |A| = -19 \\ |A_x| = -76 \\ |A_y| = 57 \end{array} \][/tex]
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