Problem 10 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Compute the indicated products. \(\left[\begin{array}{rrr}3 & 2 & -1 \\ 4 & -1 & 0 \\ -5 & 2 & 1\end{array}\right]\left[\begin{array}{r}3 \\ -2 \\ 0\end{array}\right]\)

Step-by-Step Solution

Answer

The product of the given matrices is: \(\left[\begin{array}{rrr}3 & 2 & -1 \\\ 4 & -1 & 0 \\\ -5 & 2 & 1\end{array}\right]\left[\begin{array}{r}3 \\\ -2 \\\ 0\end{array}\right] = \left[\begin{array}{r}5 \\\ 14 \\\ -19\end{array}\right]\)

Step 1: Set up the matrix multiplication

To set up the matrix multiplication, we can write the matrices as: \(A = \left[\begin{array}{rrr}3 & 2 & -1 \\\ 4 & -1 & 0 \\\ -5 & 2 & 1\end{array}\right]\) and \(B = \left[\begin{array}{r}3 \\\ -2 \\\ 0\end{array}\right]\) Then, we want to compute their product: \(C = AB\)

Step 2: Calculate the dot products

Now, for each row in matrix A, we need to compute the dot product with the column in matrix B (as there is only one column in B): 1. Dot product of first row of A and column of B: \(c_{11} = (3)(3) + (2)(-2) + (-1)(0) = 9 - 4 - 0 = 5\) 2. Dot product of second row of A and column of B: \(c_{21} = (4)(3) + (-1)(-2) + (0)(0) = 12 + 2 + 0 = 14\) 3. Dot product of third row of A and column of B: \(c_{31} = (-5)(3) + (2)(-2) + (1)(0) = -15 - 4 + 0 = -19\)

Step 3: Write the resulting product matrix

Now that we have calculated the necessary dot products, we can write the resulting 3x1 matrix: \(C = \left[\begin{array}{r}5 \\\ 14 \\\ -19\end{array}\right]\) Thus, the product of the given matrices is: \(\left[\begin{array}{rrr}3 & 2 & -1 \\\ 4 & -1 & 0 \\\ -5 & 2 & 1\end{array}\right]\left[\begin{array}{r}3 \\\ -2 \\\ 0\end{array}\right] = \left[\begin{array}{r}5 \\\ 14 \\\ -19\end{array}\right]\)

Key Concepts

Matrix Algebra

Matrix Algebra is a branch of mathematics that focuses on operations involving matrices. Matrices are rectangular arrays of numbers arranged in rows and columns. To perform operations on matrices, certain rules must be followed. A key operation in matrix algebra is matrix multiplication.
Matrix multiplication involves two matrices, often labeled as Matrix A and Matrix B. To multiply two matrices, the number of columns in the first matrix (Matrix A) must be equal to the number of rows in the second matrix (Matrix B). This compatibility requirement ensures that each element in the resulting matrix, often called Matrix C, is correctly calculated.
In practice, you multiply the elements of each row of the first matrix by the corresponding elements of each column of the second matrix, then sum these products to get a single number. This process is known as the dot product. Each entry in the resulting matrix is the dot product of a row from Matrix A and a column from Matrix B. The resulting matrix's size is determined by the number of rows from the first matrix and the number of columns from the second matrix.

Dot Product

The Dot Product is a critical concept in both matrix algebra and vector calculus. It is the multiplication of corresponding elements from a row and a column, summed together to yield a single number. In the context of matrices, the dot product is used to compute each entry of the resulting product matrix.
To find the dot product of a row vector from Matrix A and a column vector from Matrix B, you perform the following steps:

• Multiply the corresponding elements of the row and column together.
• Add the products of these multiplications together.
• The sum of these products is the resulting entry in the product matrix.

For example, suppose we have a row from Matrix A: \( [3, 2, -1] \) and a column from Matrix B: \( [3, -2, 0] \). Their dot product is calculated as \( (3 \cdot 3) + (2 \cdot -2) + (-1 \cdot 0) = 9 - 4 + 0 = 5 \). Each resulting value fills a position in the new matrix resulting from the matrix multiplication.

Linear Algebra

Linear Algebra is a vast area of mathematics focusing on vectors, vector spaces, linear transformations, and matrices. The matrix operations such as addition, subtraction, and multiplication are fundamental to many areas in mathematics and applied sciences.
In linear algebra, matrices act as representations of linear transformations. When you multiply matrices, you are essentially applying one linear transformation after another. This property makes matrices a powerful tool for solving systems of linear equations and transforming data.
Understanding the concepts of matrix multiplication and the dot product is crucial because they are foundational to many applications. For instance, in computer graphics, matrices are used to perform transformations, such as scaling and rotating images. In engineering, matrices help solve equations that describe physical phenomena. Thus, the study of matrix algebra within linear algebra provides essential tools for various scientific and engineering disciplines.