Question

consider the system of equations. find the determinant value of the matrix of the coefficients of the variables.

\\(x + y + z = 4\\)
\\(4x + 3y + 3z = 6\\)
\\(3y + 3z - 4x = 9\\)

\\(|a| = 6\\)
\\(|a| = 7\\)

Explanation:

Construct the coefficient matrix

Using the Coefficient Matrix knowledge point

$$
LATEXBLOCK0
$$

Calculate the determinant

Using the Matrix Determinant knowledge point

$$
LATEXBLOCK1
$$

Analyze the incorrect options

To find why a student might get \(|A| = 7\), we examine a common mistake where the third equation's variables are not rearranged. If the coefficients are taken directly from the written order \(3y + 3z - 4x = 9\) as \((3, 3, -4)\):

$$
LATEXBLOCK2
$$

This confirms that failing to align the variables in the third equation leads to the incorrect option \(|A| = 7\).

Answer:

• |A| = 6
• |A| = 7