Problem 27 - Algebra 2

Question

Find each product, if possible. \(\left[\begin{array}{r}{-4} \\ {8}\end{array}\right] \cdot\left[\begin{array}{ll}{-3} & {-1}\end{array}\right]\)

Step-by-Step Solution

Answer

The product is \(\begin{bmatrix} 12 & 4 \\ -24 & -8 \end{bmatrix}\).

Step 1: Understand the Matrix Dimensions

We have two matrices to multiply. The first matrix is a 2x1 column vector: \[\begin{bmatrix} -4 \ 8 \end{bmatrix}\]and the second matrix is a 1x2 row vector: \[\begin{bmatrix} -3 & -1 \end{bmatrix}\].

Step 2: Determine Feasibility of Multiplication

Matrix multiplication is feasible if the number of columns in the first matrix matches the number of rows in the second matrix. Here, the first matrix has 1 column and the second matrix has 1 row, so multiplication is possible.

Step 3: Perform the Matrix Multiplication

We will multiply the matrices by taking the dot product of the column vector and the row vector, resulting in a 2x2 matrix:\[\begin{bmatrix} -4 \ 8 \end{bmatrix} \ ext{x} \begin{bmatrix} -3 & -1 \end{bmatrix} = \begin{bmatrix} -4\times(-3) & -4\times(-1) \ 8\times(-3) & 8\times(-1) \end{bmatrix}\].

Step 4: Calculate Individual Products

Compute each of the elements in the resulting matrix:- \(-4\times(-3) = 12\)- \(-4\times(-1) = 4\)- \(8\times(-3) = -24\)- \(8\times(-1) = -8\)The resulting matrix is:\[\begin{bmatrix} 12 & 4 \ -24 & -8 \end{bmatrix}\].

Key Concepts

Matrices

Matrices are a fundamental concept used in various fields, such as mathematics, computer science, physics, and economics. They are typically organized in rows and columns, forming a rectangular array of numbers or expressions. Here's a closer look at some key aspects of matrices:

• Dimensions: The dimensions of a matrix describe its size by stating the number of rows and columns it contains. For instance, a 2x3 matrix has 2 rows and 3 columns.
• Types of Matrices: Various types exist, including square matrices (equal number of rows and columns), row vectors (single row), and column vectors (single column).
• Matrix Notation: Matrices are usually denoted with capital letters such as \( A \), \( B \), or \( C \). Elements within are often referenced using subscripts, like \( a_{mn} \), where \( m \) is the row number and \( n \) is the column number.

To solve a problem involving matrices, like our example, understanding their dimensions is crucial, as it impacts how operations such as multiplication are performed. The matrices in the given example include a column vector and a row vector, both playing a specific role in the multiplication process.

Vector Multiplication

Vector multiplication is an operation that involves combining vectors in specific ways. It can take different forms, such as scalar multiplication or dot product, each with distinct characteristics. Our exercise focuses on multiplying a column vector by a row vector, resulting in a matrix. This particular form of vector multiplication yields a matrix through these methods:

• Column Vector: A vector with multiple rows but only one column. It is used here as one element of the multiplication process.
• Row Vector: A vector with a single row and multiple columns. This complements the column vector during multiplication.
• Process: When these two types of vectors undergo multiplication, the product is a matrix. Each element of the resulting matrix is obtained by multiplying entries of the column vector by the corresponding entries of the row vector.

Vector multiplication of this kind applies principles similar to matrix multiplication but focuses on a more streamlined process due to the nature of vectors having just a single row or column. It is instrumental in simplifying complex operations and is foundational to more advanced algebraic manipulations.

Dot Product

The dot product, also known as the scalar product, is a crucial operation in vector mathematics. It is often used in matrix-related computations, such as in this exercise where a column vector is multiplied with a row vector.

• Definition: The dot product of two vectors is the sum of the products of their corresponding entries. For vectors \( \mathbf{a} = [a_1, a_2, \, \ldots, \, a_n] \) and \( \mathbf{b} = [b_1, b_2, \, \ldots, \, b_n] \), it is calculated as \( a_1b_1 + a_2b_2 + \ldots + a_nb_n \).
• Applications: The dot product can determine the angle between vectors, project one vector onto another, and check orthogonality.
• In Matrix Multiplication: When calculating matrix products, like the example, each entry of the resulting matrix is computed using the dot product of the column from one matrix and the row from another.

Within the context of the provided exercise, the dot product forms the foundation for building the resultant matrix by systematically multiplying each component of the vectors and summing the results accordingly. It highlights the interconnectedness of various mathematical concepts in producing complex solutions.