Problem 9 - Algebra and Trigonometry

Question

Perform the matrix operation, or if it is impossible, explain why. $$ 3\left[\begin{array}{rr}{1} & {2} \\ {4} & {-1} \\ {1} & {0}\end{array}\right] $$

Step-by-Step Solution

Answer

The resulting matrix is \( \begin{bmatrix} 3 & 6 \\ 12 & -3 \\ 3 & 0 \end{bmatrix} \).

Step 1: Identify the Matrix

First, identify the matrix given in the problem. The matrix is a 3x2 matrix, meaning it has 3 rows and 2 columns: \[ A = \begin{bmatrix} 1 & 2 \ 4 & -1 \ 1 & 0 \end{bmatrix} \]

Step 2: Understand Scalar Multiplication

Scalar multiplication involves multiplying each entry of the matrix by the scalar value provided. In this case, the scalar is 3.

Step 3: Multiply Each Element by the Scalar

Multiply every element of the matrix by the scalar 3. The matrix A is: \[ A = \begin{bmatrix} 1 & 2 \ 4 & -1 \ 1 & 0 \end{bmatrix} \] Multiply by 3: \[ 3 \begin{bmatrix} 1 & 2 \ 4 & -1 \ 1 & 0 \end{bmatrix} = \begin{bmatrix} 3(1) & 3(2) \ 3(4) & 3(-1) \ 3(1) & 3(0) \end{bmatrix} \]

Step 4: Complete the Multiplication

Calculate the result for each element. This gives the new matrix: \[ B = \begin{bmatrix} 3 & 6 \ 12 & -3 \ 3 & 0 \end{bmatrix} \]

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental concept in matrix operations that can be easily understood as a process of scaling. When we talk about scalar multiplication, we're referring to multiplying every element in a matrix by a constant number, called a scalar.
This operation is analogous to multiplying each entry in the matrix by the scalar separately:

• If a matrix is defined as A and you're given a scalar, say 3, then each element of matrix A must be multiplied by 3.
• For example, assume matrix A is \( \begin{bmatrix} 1 & 2 \ 4 & -1 \ 1 & 0 \end{bmatrix} \). Then multiplying by the scalar 3 means multiplying each element one by one: \( 3 \cdot 1, 3 \cdot 2, \) and so on.

This process results in a new matrix where all the elements have been scaled by the factor of the scalar, maintaining the original matrix’s shape but altering its magnitude.

3x2 Matrix

A 3x2 matrix is a specific type of rectangular array consisting of numbers arranged in rows and columns. It’s described by its dimensions, with 3 indicating the number of rows and 2 the number of columns.
Understanding these dimensions is crucial when performing matrix operations:

• Each row contains 2 elements, thereby having a consistent arrangement for calculations.
• This matrix represents a transformation where any vector it acts upon will yield results in a transformed space.

The simplicity of a 3x2 matrix makes it a common example for demonstrating matrix operations such as scalar multiplication, providing a clear view of how each element changes independently in the process.

Matrix Multiplication

Matrix multiplication is an operation that can be more complex to grasp than scalar multiplication or simply adding matrices. It involves a rule where two matrices are multiplied not by multiplying individual corresponding elements, but by applying a specific dot product method wherein rows of the first matrix are multiplied by columns of the second matrix.
Nevertheless, in this exercise, the focus is on scalar multiplication—not matrix multiplication, as it deals with a singular matrix and scalar rather than two matrices.

• If we were to consider traditional matrix multiplication, the rule to remember is that the number of columns in the first matrix must equal the number of rows in the second matrix.
• The resulting matrix's dimensions come from the outer dimensions of the multiplied matrices.

This foundational knowledge will support your understanding of broader matrix operations in algebra.

Algebra Concepts

Matrix operations, like scalar multiplication and matrix multiplication, fall under the broader discipline of linear algebra—a branch of mathematics concerned with vector spaces and the linear transformations between them.
Linear algebra encompasses various algebraic structures and concepts crucial for solving real-world problems:

• Understanding matrix operations is fundamental to linear transformations, system of equations, and many applications in fields such as computer science, engineering, and physics.
• Versatility of matrices in representing and manipulating data makes them indispensable tools in numerous technologies and computations.

Practicing these operations through exercises enhances comprehension of the intricate layers of algebra, building the foundational skills necessary for more complex applications.