Problem 68 - Algebra 1

Question

Multiply the matrix by the real number. $$\text { Sample: }-3\left[\begin{array}{rr}1 & -2 \\\\-4 & 0\end{array}\right]=\left[\begin{array}{rr}-3(1) & -3(-2) \\\\-3(-4) & -3(0)\end{array}\right]=\left[\begin{array}{rr}-3 & 6 \\\12 & 0\end{array}\right]$$ $$-8\left[\begin{array}{rr}-4 & -7 \\\3 & 3\end{array}\right]$$

Step-by-Step Solution

Answer

The result of the multiplication is \[ \begin {bmatrix} 32 & 56 \\ -24 & -24 \end {bmatrix} \].

Step 1: Analyze the given matrix and the scalar

The task is to multiply the entire matrix \[ \begin {bmatrix} -4 & -7 \\ 3 & 3 \end {bmatrix} \] by -8.

Step 2: Begin the multiplication

Multiply -8 with every element in the given matrix. This would yield: \[ -8 \times \begin {bmatrix} -4 & -7 \\ 3 & 3 \end {bmatrix} = \begin {bmatrix} -8(-4) & -8(-7) \\ -8(3) & -8(3) \end {bmatrix} \]

Step 3: Calculate the final matrix

Calculate the multiplication: \[ \begin {bmatrix} -8(-4) & -8(-7) \\ -8(3) & -8(3) \end {bmatrix} = \begin {bmatrix} 32 & 56 \\ -24 & -24 \end {bmatrix} \].

Key Concepts

Understanding Scalar Multiplication

Scalar multiplication is a fundamental concept in matrix algebra. It involves multiplying each entry of a matrix by a real number, known as a scalar. This operation is essential to scaling a matrix while maintaining its structure. For example, if you have a matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) and you multiply it by a scalar \( k \), every element of the matrix is multiplied by \( k \).

• Each element in the matrix changes by the same proportion, as determined by multiplication with the scalar.
• The size and shape of the matrix remain unchanged; only the values are altered.

This process is straightforward but crucial for various applications, such as computing dilations in transformations, where you can increase or decrease sizes systematically. Always follow the order carefully: scalar first, then each matrix element.

Exploring Basic Algebra in Matrix Operations

Algebra in the context of matrices often involves applying basic operations like addition, subtraction, and multiplication. Scalar multiplication is inherently tied to algebraic principles. Knowing how to apply these correct rules is essential for ensuring accurate results.
In our example, multiplication follows the property of distributiveness over addition. It means each element inside the brackets is treated separately, and the operation is done outside the brackets.

• When multiplying a scalar \(-8\) with a matrix \( \begin {bmatrix} -4 & -7 \ 3 & 3 \end {bmatrix} \), each element gets multiplied individually.
• The expression becomes \( -8(-4), -8(-7), -8(3), -8(3) \).
• Once calculated, we see the impact of algebraic principles in balancing changes in the matrix.

This systematic approach reflects true algebraic operations, where maintaining consistency throughout the calculation is key.

Manipulating Matrices: A Quick Overview

Matrices are arrays of numbers, systems that can represent and solve many real-world problems. In mathematics, matrices allow concise representation of complex data and transformations.
In practice, matrices are used to perform various operations, among which scalar multiplication serves as a fundamental tool. When dealing with any matrix:

• Identify its dimensions. It may be a 2x2, 3x3, or any higher order, indicating rows by columns.
• Notice how the layout can simplify calculations, especially during operations like addition and multiplication.
• Understand how each function applied to matrices, whether scaling or transforming, retains the integrity of the original data, represented within a new context.

Therefore, matrices, through themselves and operations upon them such as scalar multiplication, offer a versatile framework for various disciplines like physics, engineering, and economics.