Problem 31 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the matrix \(A\) such that $$A\left[\begin{array}{rr}1 & 0 \\\\-1 & 3\end{array}\right]=\left[\begin{array}{rr}-1 & -3 \\\3 & 6\end{array}\right]$$

Step-by-Step Solution

Answer

The short answer is: \(A = \begin{pmatrix} -2 & -1 \\ 5 & 2 \end{pmatrix}\).

Step 1: Write the matrix equation

Write down the problem as a matrix equation involving the unknown matrix A and the two given matrices: \(A\begin{pmatrix} 1 & 0 \\ -1 & 3 \end{pmatrix} = \begin{pmatrix} -1 & -3 \\ 3 & 6 \end{pmatrix}\)

Step 2: Set up the unknown matrix A

Set up the 2x2 matrix A with its elements as unknowns: \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\)

Step 3: Multiply the matrix A with the given matrix

Next, multiply the matrix A with the given matrix: \(\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 1 & 0 \\ -1 & 3 \end{pmatrix} = \begin{pmatrix} a - b & 3b \\ c - d & 3d \end{pmatrix}\)

Step 4: Equate the resulting product matrix with the given product matrix

Now, equate the elements of the resulting product matrix to the elements of the given product matrix: \(\begin{pmatrix} a - b & 3b \\ c - d & 3d \end{pmatrix} = \begin{pmatrix} -1 & -3 \\ 3 & 6 \end{pmatrix}\) From this, we can write the following system of equations: 1) \(a - b = -1\) 2) \(3b = -3\) 3) \(c - d = 3\) 4) \(3d = 6\)

Step 5: Solve the system of equations

Solve the system of equations to find the values of a, b, c, and d: From equation 2) \(b = -1\) From equation 4) \(d = 2\) Substituting the value of \(b\) in equation 1) gives: \(a - (-1) = -1\) \(a = -1+(-1) = -2\) Substituting the value of \(d\) in equation 3) gives: \(c - 2 = 3\) \(c = 3 + 2 = 5\) Thus, we now have the values for a, b, c, and d.

Step 6: Form the matrix A using the values of a, b, c, and d

Now that we have the values for a, b, c, and d, we can construct the matrix A: \(A = \begin{pmatrix} -2 & -1 \\ 5 & 2 \end{pmatrix}\) So, the matrix A that we were looking for is given by: $$A=\left[\begin{array}{rr}-2 & -1 \\\5 & 2\end{array}\right]$$

Key Concepts

System of Linear Equations

In math, a system of linear equations is a collection of two or more linear equations with the same set of variables. These systems can be interpreted as geometric lines and planes in multidimensional space, where each equation corresponds to a line in two dimensions or a plane in higher dimensions. The points of intersection of these lines or planes represent the solutions to the system. Solving a system of linear equations, such as in the provided exercise, is about finding the values for the variables that satisfy all equations simultaneously.

There are various methods to solve these systems, including substitution, elimination, and matrix methods like Gaussian elimination. In the context of this exercise, by representing the unknowns with variables within a matrix and using matrix multiplication, we are able to translate the problem into a system of equations. This allows for a straightforward process to find the values of each unknown by solving the corresponding set of linear equations.

Matrices in Mathematics

Matrices play a fundamental role in mathematics and are particularly useful in representing and solving systems of linear equations. A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Each element in the matrix can be referred to by its row and column position. Matrices are not just a convenient notation; they are powerful tools that can encapsulate complex information in a compact form. In many mathematical applications, matrices capture relationships between variables, transform geometric figures, and represent data.

In the realm of linear algebra, matrices are utilized to perform linear transformations and can represent the coefficients of a system of linear equations, which streamlines problem-solving processes such as the one demonstrated in the matrix multiplication exercise.

Matrix Algebra

Matrix algebra is a branch of mathematics dealing with the study and manipulation of matrices and the algebraic properties that govern these operations. It extends traditional algebraic concepts to a system of multiple linear equations and introduces operations such as addition, subtraction, scalar multiplication, and matrix multiplication, as shown in the textbook problem. Matrix multiplication, in particular, is a non-commutative operation that combines two matrices to produce a third matrix.

Understanding matrix multiplication is crucial, as it is not just about multiplying corresponding elements but involves the dot product of rows and columns. This process is integral to finding the solution to matrix equations and systems of linear equations. In the context of the exercise, we used matrix multiplication to relate an unknown matrix to a given product and rearranged the outcome into a system of linear equations, which we then solved to find the unknown values.