Open Question
Use Cramer's rule to solve each system of equations. If D = 0, then use another methodto determine the solution set. See Examples 5–7. x + 2y + 3z = 4 4x + 3y + 2z = 1 -x - 2y - 3z = 0
7. Systems of Equations & Matrices
Determinants and Cramer's Rule
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Evaluate the determinant of the matrix.
A
165
B
9
C
63
D
25
Verified step by step guidance1
Identify the matrix as a 3x3 matrix, which means we will use the formula for the determinant of a 3x3 matrix.
The formula for the determinant of a 3x3 matrix \( A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \) is \( ext{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \).
Substitute the values from the matrix into the formula: \( a = 4, b = 1, c = 2, d = -5, e = 2, f = -3, g = 1, h = -4, i = 6 \).
Calculate each of the three terms separately: \( 4(2 \cdot 6 - (-3) \cdot (-4)) \), \( -1((-5) \cdot 6 - 1 \cdot (-3)) \), and \( 2((-5) \cdot (-4) - 1 \cdot 2) \).
Add the results of the three terms together to find the determinant of the matrix.
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