Carletta has $10,000 to invest. As her financial consultant you recommend that she invest in Treasury Bills that yield 6%, Treasury Bonds that yield 7%, and coroporate bonds that yield 8%. Carletta wants to have an annual income of $680, and the amount invested in corporate bonds must be half that invested in Treasury Bills. Find the amound she should place in each investment using matrix.

Question

Carletta has $10,000 to invest. As her financial consultant you recommend that she invest in Treasury Bills that yield 6%, Treasury Bonds that yield 7%, and coroporate bonds that yield 8%. Carletta wants to have an annual income of $680, and the amount invested in corporate bonds must be half that invested in Treasury Bills. Find the amound she should place in each investment using matrix.

Answer 1

why use a matrix for such a simple question?

let the amount invested at corporate bonds be $x
then the amount invested in treasurey bonds is $2x
then the amount invested in treasure bills is 10000-x-2x

then .06(10000-3x) + .07(2x) + .08x = 680

solve for x

Answer 2

To solve this problem using a matrix, we can set up a system of equations. Let's define the variables as follows:

x = amount invested in Treasury Bills
y = amount invested in Treasury Bonds
z = amount invested in Corporate Bonds

Based on the given information, we can write the following equations:

Equation 1: x + y + z = $10,000 (total amount invested)
Equation 2: 0.06x + 0.07y + 0.08z = $680 (total annual income)

However, it's important to note that the amount invested in corporate bonds must be half of that invested in Treasury Bills. So we can add an additional equation:

Equation 3: z = 0.5x

Now we have a system of three equations with three variables. We can represent this system in matrix form as follows:

[1 1 1] [x] [10,000]
[0.06 0.07 0.08] [y] = [680]
[0.5 0 -1] [z] [0]

To solve this matrix equation, we can use Gaussian elimination or matrix inversion. However, since it seems like you are looking for a specific amount for each investment, we can use matrix inversion to directly find the values of x, y, and z.

First, we need to invert the coefficient matrix. Let's call it A:

A = [[1 1 1], [0.06 0.07 0.08], [0.5 0 -1]]

Next, we need to find the inverse of A, let's call it A_inv:

A_inv = inverse(A)

Finally, we can find the solution vector X by multiplying the inverse by the constant vector:

X = A_inv * [10,000 680 0]

Once we solve this multiplication, we will get the values for x, y, and z, which will represent the amounts Carletta should invest in each type of investment.