Problem 36 - Algebra 2

Question

Perform the indicated matrix operation. If the matrix does not exist, write impossible. $$ 8\left[\begin{array}{cc}{0.25} & {0.5} \\ {0.75} & {1.5}\end{array}\right]-2\left[\begin{array}{cc}{0.25} & {0.5} \\ {0.75} & {1.5}\end{array}\right] $$

Step-by-Step Solution

Answer

The result is the matrix \( \begin{bmatrix} 1.5 & 3 \\ 4.5 & 9 \end{bmatrix} \).

Step 1: Understand the Problem

We need to perform a matrix operation involving scalar multiplication and matrix subtraction. Each matrix in the operation is the same 2x2 matrix, with elements [0.25, 0.5; 0.75, 1.5].

Step 2: Perform Scalar Multiplication

Multiply the given matrix by the scalar 8. This means each element in the matrix is multiplied by 8. Thus, \( 8 \times \begin{bmatrix} 0.25 & 0.5 \ 0.75 & 1.5 \end{bmatrix} = \begin{bmatrix} 8 \times 0.25 & 8 \times 0.5 \ 8 \times 0.75 & 8 \times 1.5 \end{bmatrix} = \begin{bmatrix} 2 & 4 \ 6 & 12 \end{bmatrix} \).

Step 3: Perform Scalar Multiplication Again

Multiply the same matrix by the scalar 2. This involves the same process as before: \( 2 \times \begin{bmatrix} 0.25 & 0.5 \ 0.75 & 1.5 \end{bmatrix} = \begin{bmatrix} 2 \times 0.25 & 2 \times 0.5 \ 2 \times 0.75 & 2 \times 1.5 \end{bmatrix} = \begin{bmatrix} 0.5 & 1 \ 1.5 & 3 \end{bmatrix} \).

Step 4: Subtract the Matrices

Subtract the matrix from Step 3 from the matrix in Step 2:\(\begin{bmatrix} 2 & 4 \ 6 & 12 \end{bmatrix} - \begin{bmatrix} 0.5 & 1 \ 1.5 & 3 \end{bmatrix} = \begin{bmatrix} 2-0.5 & 4-1 \ 6-1.5 & 12-3 \end{bmatrix} = \begin{bmatrix} 1.5 & 3 \ 4.5 & 9 \end{bmatrix} \).

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental operation in matrix algebra where each element of a matrix is multiplied by a constant number, known as a scalar. This operation is quite straightforward but extremely powerful, serving numerous purposes in both pure and applied mathematics.
To perform scalar multiplication:

• Select the matrix you want to perform the operation on.
• Multiply every entry of the matrix by the given scalar.

For example, if we have a matrix \([ \begin{bmatrix} 0.25 & 0.5 \ 0.75 & 1.5 \end{bmatrix} ]\) and a scalar like 8, then the calculation involves multiplying each entry by this scalar. So, \\[8 \times \begin{bmatrix} 0.25 & 0.5 \ 0.75 & 1.5 \end{bmatrix} = \begin{bmatrix} 2 & 4 \ 6 & 12 \end{bmatrix}\]Scalar multiplication does not change the dimensions of the matrix, ensuring the result remains a 2x2 matrix when starting with one. So, you can believe in its simplicity while relying on its utility.

Matrix Subtraction

Matrix subtraction is another basic matrix operation where we subtract corresponding elements from two matrices of the same size to produce a new matrix. Key Steps in Matrix Subtraction:

• Ensure both matrices have the same dimensions. If they do not, subtraction is impossible.
• Subtract each element of the second matrix from the corresponding element of the first matrix.

For instance, given two matrices:
\[\begin{bmatrix} 2 & 4 \ 6 & 12 \end{bmatrix} \quad \text{and} \quad \begin{bmatrix} 0.5 & 1 \ 1.5 & 3 \end{bmatrix}\]the subtraction is performed as follows:
\[\begin{bmatrix} 2 - 0.5 & 4 - 1 \ 6 - 1.5 & 12 - 3 \end{bmatrix} = \begin{bmatrix} 1.5 & 3 \ 4.5 & 9 \end{bmatrix}\] It's simple but critical to verify that the matrices are compatible for subtraction, preserving consistency in matrix operations.

2x2 Matrix

A 2x2 matrix is a simple square matrix with two rows and two columns, making it an important element in linear algebra due to its manageability and widespread applicability in various fields.
2x2 matrices are often used for:

• Solving systems of linear equations.
• Transformations in geometry, such as rotations or scaling.
• Modeling real-world phenomena in economics or physics.

Given:\[\begin{bmatrix} a & b \ c & d \end{bmatrix} \]this matrix has four elements you can reference directly. They are often the building blocks for more complex structures in higher dimensions, yet they embody vital properties and usefulness.
When working with operations like scalar multiplication or subtraction on such matrices, their small size helps clarify results quickly and allows easy illustration of concepts without the potential complexity of larger matrices.