Question

An electronics company produces​ transistors, resistors, and computer chips. Each transistor requires 3 units of​ copper, 1 unit of​ zinc, and 2 units of glass. Each resistor requires 3​, 2​, and 1 units of the three​ materials, and each computer chip requires 2​, 1​, and 2 units of these​ materials, respectively. How many of each product can be made with 1500 units of​ copper, 780 units of​ zinc, and 1110 units of​ glass? Solve this exercise by using the inverse of the coefficient matrix to solve a system of equations.
The company can make
​transistors,
​resistors, and
computer chips.

Answer 1

Answer to Question #89249 – Math – Linear Algebra

Question

An electronics company produces transistors, resistors, and computer chips. Each transistor requires 3 units of copper, 1 unit of zinc, and 2 units of glass. Each resistor requires 3, 2, and 1 units of the three materials, and each computer chip requires 2, 1, and 2 units of these materials, respectively. How many of each product can be made with 1500 units of copper, 780 units of zinc, and 1110 units of glass? Solve this exercise by using the inverse of the coefficient matrix to solve a system of equations.

The company can make transistors, resistors, and computer chips.

Solution

Let the number of Transisters = x.

Let the number of Resisters = y.

Let the number of Computer Chips = z.

Given the each transistor requires 3 units of copper, 1 units of zinc, and 2 units of glass.

Given that each resistor requires 3 units of copper, 2 units of zinc, and 1 unit of glass.

Given that each computer chip requires 2 units of copper, 1 unit of zinc, and 2 units of glass.

Total number of copper units = 1500.

Total number of zinc units = 780.

Total number of glass units = 1110.

We can form the linear equations in the following way:

3(Transistors)+3(resisters)+2(computer chips)=total number of copper units

$$\Rightarrow 3x + 3y + 2z = 1500 \dots \dots (1)$$

2(Transistors)+2(resisters)+1(computer chip)=total number of zinc units

$$\Rightarrow x + 2y + 1z = 780 \dots \dots (2)$$

2(Transistors)+1(resisters)+2(computer chip)=total number of glass units

$$\Rightarrow 2x + y + 2z = 1110 \dots \dots (3)$$

Equations (1), (2) and (3) can be written in the matrix notation as follows:

$$\left[ \begin{array}{ccc} 3 & 3 & 2 \\ 1 & 2 & 1 \\ 2 & 1 & 2 \end{array} \right] \left[ \begin{array}{c} x \\ y \\ z \end{array} \right] = \left[ \begin{array}{c} 1500 \\ 780 \\ 1110 \end{array} \right] \dots \dots (4)$$

To find $\left[ \begin{array}{l}x\\ y\\ z \end{array} \right]$ we use matrix inverse method.

From (4), $\begin{bmatrix} x\\ y\\ z \end{bmatrix} = \begin{bmatrix} 3 & 3 & 2\\ 1 & 2 & 1\\ 2 & 1 & 2 \end{bmatrix}^{-1}\begin{bmatrix}1500\\ 780\\ 1110 \end{bmatrix}\dots \dots$ (5)

Let the coefficient matrix be $A = \begin{bmatrix} 3 & 3 & 2 \\ 1 & 2 & 1 \\ 2 & 1 & 2 \end{bmatrix}$

Determinant of $A = \begin{vmatrix} 3 & 3 & 2 \\ 1 & 2 & 1 \\ 2 & 1 & 2 \end{vmatrix}$

Expanding the determinant we get $|A| = 3\left| \begin{array}{cc}2 & 1\\ 1 & 2 \end{array} \right| - 3\left| \begin{array}{cc}1 & 1\\ 2 & 2 \end{array} \right| + 2\left| \begin{array}{cc}1 & 2\\ 2 & 1 \end{array} \right|$

$$\begin{array}{l} = 3(4 - 1) - 3(2 - 2) + 2(1 - 4) \quad (\text{Since}, \left| \begin{array}{cc} a & b \\ c & d \end{array} \right| = ad - bc) \\ = 3(3) - 3(0) + 2(-3) \\ = 9 - 0 - 6 \\ = 3 \\ \end{array}$$

Therefore, Determinant of A is $|A| = 3 \neq 0$

$A^{-1}$ exists.

Cofactors of an element $a_{ij}$ of A are denoted by $A_{ij}$

$$\begin{array}{l} A_{11} = (-1)^{1+1} \left| \begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array} \right| \\ = (4 - 1) \\ = 3 \\ \end{array}$$ $$\begin{array}{l} A_{12} = (-1)^{1+2} \left| \begin{array}{cc} 1 & 1 \\ 2 & 2 \end{array} \right| \\ = -(4 - 4) \\ = 0 \\ \end{array}$$ $$\begin{array}{l} A_{13} = (-1)^{1+3} \left| \begin{array}{cc} 1 & 2 \\ 2 & 1 \end{array} \right| \\ = (1 - 4) \\ = -3 \\ \end{array}$$ $$\begin{array}{l} A_{21} = (-1)^{2+1} \left| \begin{array}{cc} 3 & 2 \\ 1 & 2 \end{array} \right| \\ = -(6 - 2) \\ = -4 \\ \end{array}$$ $$\begin{array}{l} A_{22} = (-1)^{2+2} \left| \begin{array}{cc} 3 & 2 \\ 2 & 2 \end{array} \right| \\ = (6 - 4) \\ = 2 \\ \end{array}$$ $$\begin{array}{l} A_{23} = (-1)^{2+3} \left| \begin{array}{cc} 3 & 3 \\ 2 & 1 \end{array} \right| \\ = -(3 - 6) \\ = 3 \\ \end{array}$$ $$\begin{array}{l} A_{31} = (-1)^{3+1} \left| \begin{array}{cc} 3 & 2 \\ 2 & 1 \end{array} \right| \\ = (3 - 4) \\ = -1 \\ \end{array}$$ $$\begin{array}{l} A_{32} = (-1)^{3+2} \left| \begin{array}{cc} 3 & 2 \\ 1 & 1 \end{array} \right| \\ = -(3 - 2) \\ = -1 \\ \end{array}$$ $$\begin{array}{l} A_{33} = (-1)^{3+3} \left| \begin{array}{cc} 3 & 3 \\ 1 & 2 \end{array} \right| \\ = (6 - 3) \\ = 3 \\ \end{array}$$

The cofactors matrix is = $\begin{bmatrix} 3 & 0 & -3 \\ -4 & 2 & 3 \\ -1 & -1 & 3 \end{bmatrix}$

Adjoint A = Adj A = Transpose of cofactors matrix

$$\begin{array}{l} = \left[ \begin{array}{ccc} 3 & 0 & -3 \\ -4 & 2 & 3 \\ -1 & -1 & 3 \end{array} \right]^T \\ \left[ \begin{array}{ccc} 3 & -4 & -1 \\ 0 & 2 & -1 \\ -3 & 3 & 3 \end{array} \right] \\ \end{array}$$ $$\begin{array}{l} \text{Therfore, Adj}(A) = \left[ \begin{array}{ccc} 3 & -4 & -1 \\ 0 & 2 & -1 \\ -3 & 3 & 3 \end{array} \right] \\ A^{-1} = \frac{Adj(A)}{\det A} \\ = \frac{1}{3} \left[ \begin{array}{ccc} 3 & -4 & -1 \\ 0 & 2 & -1 \\ -3 & 3 & 3 \end{array} \right] \\ \end{array}$$ $$\begin{array}{l} \text{Therefore, } A^{-1} = \frac{1}{3} \left[ \begin{array}{ccc} 3 & -4 & -1 \\ 0 & 2 & -1 \\ -3 & 3 & 3 \end{array} \right] \\ \end{array}$$

Substituting $A^{-1}$, in (5) we get $\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \frac{1}{3} \begin{bmatrix} 3 & -4 & -1 \\ 0 & 2 & -1 \\ -3 & 3 & 3 \end{bmatrix} \begin{bmatrix} 1500 \\ 780 \\ 1110 \end{bmatrix}$

$$\begin{array}{l} = \frac{1}{3} \left[ \begin{array}{c} 3(1500) - 4(780) - 1110 \\ 0(1500) + 2(780) - 1110 \\ -3(1500) + 3(780) + 3(1110) \end{array} \right] \\ = \frac{1}{3} \left[ \begin{array}{c} 270 \\ 450 \\ 1170 \end{array} \right] \\ = \left[ \begin{array}{c} 90 \\ 150 \\ 390 \end{array} \right] \\ \end{array}$$

Therefore, $\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 90 \\ 150 \\ 390 \end{bmatrix}$, which the company produces 90 transistors, 150 resistors, and 390 computer chip.

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