Problem 32 - Algebra 2

Question

Mental Math Find each product. $$ 0.5\left[\begin{array}{cc}{3} & {14} \\ {7} & {-4}\end{array}\right] $$

Step-by-Step Solution

Answer

[[1.5, 7], [3.5, -2]]

Step 1: Understanding the Exercise

Firstly, understand that we need to find the product of the scalar value 0.5 and the matrix. The matrix has four elements, and each element needs to be multiplied by 0.5.

Step 2: Multiplying the Matrix by the Scalar

Multiply each element in the matrix by 0.5. Perform the operation independently for each element: For element (1,1): 0.5 * 3 = 1.5 For element (1,2): 0.5 * 14 = 7 For element (2,1): 0.5 * 7 = 3.5 For element (2,2): 0.5 * (-4) = -2

Key Concepts

Scalar Multiplication

Scalar multiplication is an operation in which every element within a matrix is multiplied by a fixed number, known as a 'scalar.' In the context of our exercise, the scalar is 0.5. Performing scalar multiplication is straightforward: each entry in the matrix must be multiplied by this value.

Think of it as resizing the matrix in accordance to the value of the scalar. If the scalar is greater than 1, the matrix grows in magnitude. Conversely, if the scalar is between 0 and 1, like in our exercise, the matrix elements reduce in size. Scalar multiplication is essential as it forms the basis for more complex matrix operations.

Matrix Operations

Matrix operations are the procedures that can be performed on matrices, including addition, subtraction, multiplication, and scalar multiplication, among others. These operations are fundamental components of linear algebra and are widely used in various fields like physics, engineering, and computer science.

For scalar multiplication, each element in a matrix is multiplied by the same scalar. Unlike matrix-matrix multiplication, scalar multiplication doesn't require that matrices conform to size restrictions. It's one of the simplest matrix operations, yet core to understanding more complex algebraic expressions involving matrices.

Mental Math

Mental math refers to performing arithmetic calculations in one's head without the use of calculators or paper. It's a skill that can be developed with practice and is incredibly useful in many scenarios, including performing quick calculations in everyday life or getting a rough estimate before using more precise tools.

In our given exercise, mental math can be applied to quickly determine the result of scaling our matrix by 0.5. For instance, halving each element is an intuitive task: half of 3 is 1.5, half of 14 is 7, and so on. Developing strong mental math skills can not only speed up such calculations but also deepen your understanding of numerical relationships.

Algebraic Expressions

Algebraic expressions are mathematical phrases containing numbers, variables, and operation symbols. When working with matrices, scalar multiplication introduces algebraic expressions involving both matrices and scalars. These expressions follow the standard algebraic rules but are extended to matrix elements.

In the context of the exercise, when we scale the matrix by 0.5, we are essentially creating new algebraic expressions for each element of the matrix. For example, the expression for the element in the first row, first column becomes \(0.5 \times 3\) which simplifies to \(1.5\). These expressions reveal the relationships between the original matrix and the scalar, resulting in a new matrix after the operation.