What is the determinant of the coefficient matrix of the system $\left\{\begin{array}{l}-x-y+2 z=5 \\ 3 x+2 y-z=3 \\ 4 x+4 y-8 z=-2\end{array}\right.\$?
A. 0
B. 16
C. 18
D. 40
Answer (2)
Write down the coefficient matrix of the system of equations.
Apply the formula for calculating the determinant of a 3x3 matrix.
Simplify the expression to find the determinant.
The determinant of the coefficient matrix is 0 .
Explanation
Identifying the Coefficient Matrix We are asked to find the determinant of the coefficient matrix of the given system of equations. The system is:
⎩ ⎨ ⎧ − x − y + 2 z = 5 3 x + 2 y − z = 3 4 x + 4 y − 8 z = − 2
The coefficient matrix is formed by the coefficients of the variables x , y , and z in each equation.
Writing the Coefficient Matrix The coefficient matrix A is:
A = − 1 3 4 − 1 2 4 2 − 1 − 8
Applying the Determinant Formula To find the determinant of a 3 × 3 matrix, we use the following formula:
a d g b e h c f i = a ( e i − f h ) − b ( d i − f g ) + c ( d h − e g )
Applying this to our matrix A :
− 1 3 4 − 1 2 4 2 − 1 − 8 = − 1 ( 2 × ( − 8 ) − ( − 1 ) × 4 ) − ( − 1 ) ( 3 × ( − 8 ) − ( − 1 ) × 4 ) + 2 ( 3 × 4 − 2 × 4 )
Calculating the Determinant Now, we simplify the expression:
= − 1 ( − 16 + 4 ) + 1 ( − 24 + 4 ) + 2 ( 12 − 8 )
= − 1 ( − 12 ) + 1 ( − 20 ) + 2 ( 4 )
= 12 − 20 + 8
= 0
Final Answer Therefore, the determinant of the coefficient matrix is 0.
Examples
The determinant of a coefficient matrix can tell us about the solutions to a system of linear equations. If the determinant is non-zero, the system has a unique solution. If the determinant is zero, the system either has no solution or infinitely many solutions. This concept is used in various fields like engineering, economics, and computer science to analyze and solve systems of equations.
The determinant of the coefficient matrix of the given system of equations is calculated to be 0. This indicates that the system does not have a unique solution. Therefore, the answer is A. 0.
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