Problem 16 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Compute the indicated products. \(\left[\begin{array}{ll}1.2 & 0.3 \\ 0.4 & 0.5\end{array}\right]\left[\begin{array}{rr}0.2 & 0.6 \\ 0.4 & -0.5\end{array}\right]\)

Step-by-Step Solution

Answer

The product of the given matrices is: \(\left[\begin{array}{ll}1.2 & 0.3 \\\ 0.4 & 0.5\end{array}\right]\left[\begin{array}{rr}0.2 & 0.6 \\\ 0.4 & -0.5\end{array}\right] = \left[\begin{array}{rr}0.36 & 0.57 \\\ 0.28 & -0.01\end{array}\right]\).

Step 1: Understand how to multiply matrices

When multiplying two matrices, we use the following formula for the entry located in the i-th row and j-th column of the resulting matrix: \((AB)_{ij} = \sum_{k=1}^n A_{ik}B_{kj}\), where n is the number of columns in matrix A (which should be equal to the number of rows in matrix B). Consider the given matrices: \(A = \left[\begin{array}{ll}1.2 & 0.3 \\\ 0.4 & 0.5\end{array}\right]\) and \(B = \left[\begin{array}{rr}0.2 & 0.6 \\\ 0.4 & -0.5\end{array}\right]\).

Step 2: Compute the elements of the product matrix

We will use the matrix multiplication formula to compute each element of the resulting 2x2 matrix. 1. Compute \((AB)_{11}\): \((AB)_{11} = A_{11}B_{11} + A_{12}B_{21} = (1.2)(0.2) + (0.3)(0.4) = 0.24 + 0.12 = 0.36\) 2. Compute \((AB)_{12}\): \((AB)_{12} = A_{11}B_{12} + A_{12}B_{22} = (1.2)(0.6) + (0.3)(-0.5) = 0.72 - 0.15 = 0.57\) 3. Compute \((AB)_{21}\): \((AB)_{21} = A_{21}B_{11} + A_{22}B_{21} = (0.4)(0.2) + (0.5)(0.4) = 0.08 + 0.20 = 0.28\) 4. Compute \((AB)_{22}\): \((AB)_{22} = A_{21}B_{12} + A_{22}B_{22} = (0.4)(0.6) + (0.5)(-0.5) = 0.24 - 0.25 = -0.01\)

Step 3: Write the resulting product matrix

Now that we have all the elements of the product matrix, let's write the resulting 2x2 matrix: \[AB = \left[\begin{array}{rr}0.36 & 0.57 \\\ 0.28 & -0.01\end{array}\right]\] So, the product of the given matrices is: \(\left[\begin{array}{ll}1.2 & 0.3 \\\ 0.4 & 0.5\end{array}\right]\left[\begin{array}{rr}0.2 & 0.6 \\\ 0.4 & -0.5\end{array}\right] = \left[\begin{array}{rr}0.36 & 0.57 \\\ 0.28 & -0.01\end{array}\right]\).

Key Concepts

Matrices

Matrices are essentially rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. The power of matrices lies in their ability to encode information succinctly and to allow operations such as addition, multiplication, and inversion.

They are used in various fields from computer graphics to economics, to solve linear equations and to represent complex structures in a compact form. In the context of our exercise, two matrices are given: matrix A and matrix B, each of size 2x2.

In general, matrices offer a systematic way to handle and manipulate data. The number specified at each position in a matrix is known as an entry or element. The knowledge of matrix operations is essential in advancing through linear algebra.

Linear Algebra

Linear algebra is a branch of mathematics that is focused on vector spaces and linear mappings between these spaces. It involves the study of vectors, matrices, and linear transformations.

In our exercise, we apply linear algebra to perform matrix multiplication, where the product of two matrices gives another matrix. This operation is fundamental in linear algebra, as it reveals many properties about the transformations these matrices can perform.

• Matrix multiplication is not just a numerical operation but also an abstract concept representing combinations of transformations.
• It requires a specific rule to multiply matrices: the number of columns in the first matrix must be equal to the number of rows in the second matrix.

Understanding linear algebra concepts is crucial for working with matrices efficiently.

Mathematics Education

In mathematics education, understanding the theory and application of matrices is vital. Teaching how to handle matrices helps students appreciate how numbers can be organized and manipulated efficiently in practical scenarios.

Matrix operations, like multiplication, improve computational skills and enable students to tackle more complex systems in mathematics and beyond.

• Students learn to systematically work through mathematical problems and develop logical thinking.
• It also prepares them for more advanced topics in math, such as calculus, physics, and engineering.

By practicing matrix multiplication and other operations, students build a solid foundation in mathematical reasoning and problem-solving, which are crucial skills in many professional fields.