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2 -37
D= 2 -6 1 is the coefficient matrix of a system of linear equations and D₂ =
4 2 5
What is the solution of the system of linear equations represented by D and Dx?
O A.
(5, 4, -11)
OB.
(5, 7.-9)
OC. (6,7,-11)
O D. (6, 4,-9)
46 2 -37
-41 -6 1
5
L-11 2

1 Answer

4 votes

Answer:

Explanation:

Let's denote the matrix \( D \) as follows:

\[ D = \begin{bmatrix} 2 & -6 & 1 \\ 4 & 2 & 5 \\ -11 & 46 & -37 \end{bmatrix} \]

Also, let \( D_2 \) be the matrix obtained by replacing the second column of \( D \) with the column vector \( \begin{bmatrix} 4 \\ -6 \\ 2 \end{bmatrix} \):

\[ D_2 = \begin{bmatrix} 2 & 4 & 1 \\ 4 & -6 & 5 \\ -11 & 2 & -37 \end{bmatrix} \]

Now, let's calculate the determinant \( D \) (denoted as \( \text{det}(D) \)) and \( D_2 \) (denoted as \( \text{det}(D_2) \)).

\[ \text{det}(D) = 2 \cdot (2 \cdot (-37) - 5 \cdot 46) - (-6) \cdot (4 \cdot (-37) - 5 \cdot (-11)) + 1 \cdot (4 \cdot 46 - 2 \cdot (-11)) \]

Simplify the above expression to find \( \text{det}(D) \).

\[ \text{det}(D_2) = 2 \cdot (-6 \cdot (-37) - 5 \cdot 2) - 4 \cdot (4 \cdot (-37) - 5 \cdot (-11)) + 1 \cdot (4 \cdot 2 - (-6) \cdot (-11)) \]

Simplify the above expression to find \( \text{det}(D_2) \).

If \( \text{det}(D) \\eq 0 \), then the system of equations represented by \( D \) has a unique solution, and the solution can be found by using Cramer's rule:

\[ x = \frac{\text{det}(D_1)}{\text{det}(D)} \]

\[ y = \frac{\text{det}(D_2)}{\text{det}(D)} \]

\[ z = \frac{\text{det}(D_3)}{\text{det}(D)} \]

where \( D_1 \), \( D_2 \), and \( D_3 \) are matrices obtained by replacing the first, second, and third columns of \( D \) with the column vector on the right side, respectively.

If \( \text{det}(D) = 0 \), then the system may have no solution or infinitely many solutions.

Perform the calculations, and if you find \( \text{det}(D) \\eq 0 \), you can proceed to find the values of \( x \), \( y \), and \( z \). If \( \text{det}(D) = 0 \), then the system may have no unique solution.

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