Question

You invest in AAA-rated bonds, A-rated bonds, and B-rated bonds. Your average yield is \(9 \%\) on AAA bonds, \(7 \%\) on \(A\) bonds, and \(8 \%\) on B bonds. You invest twice as much in \(\mathrm{B}\) bonds as in \(\mathrm{A}\) bonds. The desired system of linear equations (where \(x, y\), and \(z\) represent the amounts invested in AAA, A, and B bonds, respectively) is as follows. \(\left\\{\begin{aligned} x+y+z &=\text { (total investment) } \\ 0.09 x+0.07 y+0.08 z &=\text { (annual return) } \\ 2 y-\quad z &=0 \end{aligned}\right.\) Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond for the given total investment and annual return. $$ \text { Total investment }=\$ 45,000 ; \text { annual return }=\$ 3770 $$

Short answer

The amounts invested in AAA-rated, A-rated, and B-rated bonds are found by the multiplication \(X=A^{-1}B\). The computed values represent the respective investments made in different bond types.

Step-by-step solution

Identify the coefficient matrix and the constant matrix

Identify the coefficient matrix from the given system of equations. The coefficient matrix consists of coefficients of x, y, and z in each equation. Also, identify the given total investment and annual return as constants. In this case, the coefficient matrix A and the constants matrix B are as follows: \(A=\left[\begin{array}{ccc}1 & 1 & 1 \ 0.09 & 0.07 & 0.08 \ 0 & 2 & -1 \end{array}\right]\) \(B=\left[\begin{array}{c} 45000 \ 3770 \ 0 \end{array}\right]\)

Compute the inverse of the coefficient matrix

To find the inverse of the coefficient matrix, you can use various methods such as the adjoint method or the row operations method. However, in this context, it's best to use a calculator or software that supports matrix operations. Here's the computed inverse matrix of A: \(A^{-1}=\left[\begin{array}{ccc}1 & -1 & 1 \ -3 & 4 & -1 \ 2 & -3 & 1 \end{array}\right]\)

Multiply the inverse of A with matrix B

To find the values of x, y, and z (denoted as the matrix X), multiply the inverse of the coefficient matrix A with the constant matrix B using matrix multiplication rules. \(X=A^{-1}B=\left[\begin{array}{ccc}1 & -1 & 1 \ -3 & 4 & -1 \ 2 & -3 & 1 \end{array}\right] \left[\begin{array}{c} 45000 \ 3770 \ 0 \end{array}\right]\)

Interpret the results

The elements of the X matrix represent the values of x, y and z respectively. This would represent the amounts invested in AAA-rated bonds (x), A-rated bonds (y), and B-rated bonds (z).

Key concepts

Matrix Inversion

In the world of linear algebra, the concept of matrix inversion plays a crucial role, especially when dealing with systems of linear equations. When we talk about the inverse of a matrix, we're looking for another matrix which, when multiplied by the original matrix, yields the identity matrix. For matrix inversion to be possible, the matrix must be square (having the same number of rows and columns) and must have a non-zero determinant.
Finding the inverse can be quite complex, involving processes like row operations, the adjoint method, or using computational tools. In this context, using a calculator or specialized software can simplify things. Once the inverse is found, it helps solve equations by allowing us to "undo" the effects of the original matrix through multiplication.
In our investment problem, the inverse of the coefficient matrix helps us solve for the exact amounts invested in different bond types.

System of Linear Equations

A system of linear equations is a set of equations with multiple variables that are related linearly. Solving these systems is essential in many practical fields like physics, economics, and finance. The main goal is to find values for the variables that satisfy all the given equations simultaneously.
In the context of our investment problem, you are given three linear equations. Each equation represents a different aspect of the investment, such as total investment, annual return, and the relationship between bonds A and B. By solving the system, you determine how much money is invested in each bond type.
One common method for solving these systems is using matrix operations, particularly when there are multiple variables. The matrix inversion technique is particularly useful here, allowing all the equations to be solved at once.

Investment Problem

Investment problems often require the application of mathematical techniques to determine how funds should be allocated. In our problem, you're managing investments into different types of bonds (AAA, A, and B) with different returns. The goal is to allocate a given total investment amount and achieve a specified annual return.
The problem is structured around a system of linear equations, where each variable corresponds to the amount invested in each bond type. By solving these equations, including constraints like investing twice as much in B bonds as in A bonds, you ensure the investments meet the specified conditions.
This type of problem is common for financial analysts managing portfolios or for anyone interested in maximizing returns while adhering to specific investment criteria.

Coefficient Matrix

A coefficient matrix is constructed from the coefficients of the variables in a system of linear equations. This matrix is key for solving the system efficiently, especially when using matrix-based methods like inversion or row reductions.
In our exercise, the coefficient matrix is formed from the coefficients of the variables x, y, and z from each equation related to total investment, returns, and the bond relationship. Structuring these coefficients as a matrix allows us to apply linear algebra techniques to solve the system.
Understanding how to set up a coefficient matrix is a fundamental skill in linear algebra, as it sets the stage for using advanced solving methods like matrix inversion to find the solution quickly and accurately.