To determine the correct system of equations corresponding to the given matrix multiplication operation, we need to interpret the equation:
[tex]$
\left[\begin{array}{ccc}
5 & 2 & 1 \\
7 & -5 & 2 \\
-5 & 3 & 1
\end{array}\right] \times\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{c}
16 \\
3 \\
12
\end{array}\right]
$[/tex]
This matrix equation can be written as a system of linear equations by performing the matrix multiplication:
1. Multiply the first row of the coefficient matrix by the variable column vector:
[tex]\[ 5x + 2y + z = 16 \][/tex]
2. Multiply the second row of the coefficient matrix by the variable column vector:
[tex]\[ 7x - 5y + 2z = 3 \][/tex]
3. Multiply the third row of the coefficient matrix by the variable column vector:
[tex]\[ -5x + 3y + z = 12 \][/tex]
Thus, the system of equations corresponding to the given matrix multiplication is:
[tex]\[
\begin{cases}
5x + 2y + z = 16 \\
7x - 5y + 2z = 3 \\
-5x + 3y + z = 12
\end{cases}
\][/tex]
Comparing this system to the options given:
A. [tex]\( 5x + 2x + x = 16 \; ; \; 7y - 5y + 2y = 3 \; ; \; -5z + 3z + z = 12 \)[/tex]
B. [tex]\( 5x + 2y + z = 16 \; ; \; 7x - 5y + 2z = 3 \; ; \; -5x + 3y + z = 12 \)[/tex]
C. [tex]\( 5x + 7y - 5z = 16 \; ; \; 2x - 5y + 3z = 3 \; ; \; x + 2y + z = 12 \)[/tex]
D. [tex]\( 5x + 7y - 5z = 12 \; ; \; 2x - 5y + 3z = 3 \; ; \; x + 2y + z = 16 \)[/tex]
The correct answer is:
B. [tex]\( 5x + 2y + z = 16 ; 7x - 5y + 2z = 3 ; -5x + 3y + z = 12 \)[/tex]
