Respuesta :
Answer:
475 transistors, 25 resistors and 50 computer chips can be produced.
Step-by-step explanation:
Let us consider, p = Number of transistors.
q = Number of resistors.
r = Number of computer chips.
The following three linear equations according to question,
[tex]3\times p + 3\times q + 2\times r = 1600\\2\times p + 1\times q + 1\times r = 1025\\1\times p + 2\times q + 2\times r = 625[/tex]
The matrix form of any system, Ax = B
Where, A = Coefficient matrix
B = Constant vector
x = Variable vector
[tex]A = \left[\begin{array}{ccc}3&3&2\\2&1&1\\1&2&2\end{array}\right], x = \left[\begin{array}{ccc}p\\q\\r\end{array}\right], B = \left[\begin{array}{ccc}1600\\1025\\625\end{array}\right][/tex]
The inverse matrix, [tex]A^{-1}[/tex] can be found by using the following formula,
[tex]A^{-1} = \frac{1}{det A}\times (C_{A}) ^{T}[/tex]
Where, det A = Determinant of matrix A.
[tex]C_{A}[/tex] = Matrix of cofactors of A
Now, applying this formula to find [tex]A^{-1}[/tex];
[tex]det A = \left[\begin{array}{ccc}3&3&2\\2&1&1\\1&2&2\end{array}\right] = 3\times(2-2)-3\times(4-1)+2\times(4-1) = -3[/tex]
Here, [tex]det A\neq 0[/tex], thus the matrix is invertible.
[tex]C_{A} = \left[\begin{array}{ccc}(2-2)&-(4-1)&(4-1)\\-(6-4)&(6-2)&-(6-3)\\(3-2)&-(3-4)&(3-6)\end{array}\right] = \left[\begin{array}{ccc}0&-3&3\\-2&4&-3\\1&1&-3\end{array}\right] \\(C_{A}) ^{T} = \left[\begin{array}{ccc}0&-3&3\\-2&4&-3\\1&1&-3\end{array}\right] ^{T} = \left[\begin{array}{ccc}0&-2&1\\-3&4&1\\3&-3&-3\end{array}\right][/tex]
[tex]A^{-1} = \frac{1}{-3}\times\left[\begin{array}{ccc}0&-2&1\\-3&4&1\\3&-3&-3\end{array}\right] \\ So, x= \frac{1}{-3} \left[\begin{array}{ccc}0&-2&1\\-3&4&1\\3&-3&-3\end{array}\right]\times\left[\begin{array}{ccc}1600\\1025\\625\end{array}\right]= \frac{1}{-3} \left[\begin{array}{ccc}-1425\\-75\\-150\end{array}\right] = \left[\begin{array}{ccc}475\\25\\50\end{array}\right][/tex]
So, p = 475, q = 25, r = 50.
