Performing the Indicated Operations

Oct 13, 2025 by magicalceylontours.com 39 views

Introduction

Matrix operations are a fundamental concept in linear algebra, and they have numerous applications in various fields, including physics, engineering, and computer science. In this article, we will focus on performing matrix operations, specifically subtraction, and provide a step-by-step guide on how to do it.

What are Matrices?

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrices are used to represent systems of linear equations, and they can be added, subtracted, multiplied, and divided, just like numbers.

Matrix Subtraction

Matrix subtraction is a fundamental operation in linear algebra, and it involves subtracting one matrix from another. The resulting matrix will have the same dimensions as the original matrices.

Performing the Indicated Operations

Let's perform the indicated operations on the given matrices:

{ \left[\begin{array}{cc} 2 & -3 \\ 1 & 4 \end{array}\right] - 2 \left[\begin{array}{cc} 5 & 1 \\ 4 & 0 \end{array}\right] \}

To perform the subtraction, we need to follow these steps:

Subtract the corresponding elements: We need to subtract the corresponding elements of the two matrices. In this case, we need to subtract 2 times the elements of the second matrix from the elements of the first matrix.
Perform the scalar multiplication: We need to perform the scalar multiplication of 2 with the elements of the second matrix.
Subtract the resulting matrices: We need to subtract the resulting matrix from the first matrix.

Step 1: Subtract the Corresponding Elements

To subtract the corresponding elements, we need to subtract 2 times the elements of the second matrix from the elements of the first matrix.

{ \left[\begin{array}{cc} 2 & -3 \\ 1 & 4 \end{array}\right] - 2 \left[\begin{array}{cc} 5 & 1 \\ 4 & 0 \end{array}\right] \}

$= \left[\begin{array}{cc} 2 - 2(5) & -3 - 2(1) \\ 1 - 2(4) & 4 - 2(0) \end{array}\right]$

$= \left[\begin{array}{cc} 2 - 10 & -3 - 2 \\ 1 - 8 & 4 - 0 \end{array}\right]$

$= \left[\begin{array}{cc} -8 & -5 \\ -7 & 4 \end{array}\right]$

Step 2: Perform the Scalar Multiplication

To perform the scalar multiplication, we need to multiply the elements of the second matrix by 2.

{ 2 \left[\begin{array}{cc} 5 & 1 \\ 4 & 0 \end{array}\right] \}

$= \left[\begin{array}{cc} 2(5) & 2(1) \\ 2(4) & 2(0) \end{array}\right]$

$= \left[\begin{array}{cc} 10 & 2 \\ 8 & 0 \end{array}\right]$

Step 3: Subtract the Resulting Matrices

To subtract the resulting matrices, we need to subtract the second matrix from the first matrix.

{ \left[\begin{array}{cc} 2 & -3 \\ 1 & 4 \end{array}\right] - \left[\begin{array}{cc} 10 & 2 \\ 8 & 0 \end{array}\right] \}

$= \left[\begin{array}{cc} 2 - 10 & -3 - 2 \\ 1 - 8 & 4 - 0 \end{array}\right]$

$= \left[\begin{array}{cc} -8 & -5 \\ -7 & 4 \end{array}\right]$

Conclusion

In this article, we performed the indicated operations on the given matrices. We subtracted the corresponding elements, performed the scalar multiplication, and subtracted the resulting matrices. The resulting matrix is:

{ \left[\begin{array}{cc} -8 & -5 \\ -7 & 4 \end{array}\right] \}

This is the final answer to the problem.

Discussion

The problem involves performing matrix operations, specifically subtraction. The resulting matrix is a 2x2 matrix with elements -8, -5, -7, and 4. This matrix can be used to represent a system of linear equations.

Common Mistakes

When performing matrix operations, it's easy to make mistakes. Here are some common mistakes to avoid:

Incorrect scalar multiplication: Make sure to multiply the elements of the matrix by the correct scalar.
Incorrect subtraction: Make sure to subtract the corresponding elements of the matrices.
Incorrect resulting matrix: Make sure to get the correct resulting matrix after performing the operations.

Real-World Applications

Matrix operations have numerous real-world applications, including:

Physics: Matrix operations are used to describe the motion of objects in space.
Engineering: Matrix operations are used to design and analyze complex systems.
Computer Science: Matrix operations are used in machine learning and data analysis.

Conclusion

Introduction

Matrix operations are a fundamental concept in linear algebra, and they have numerous applications in various fields. In this article, we will answer some frequently asked questions about matrix operations.

Q: What is a matrix?

A: A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrices are used to represent systems of linear equations, and they can be added, subtracted, multiplied, and divided, just like numbers.

Q: What are the basic operations on matrices?

A: The basic operations on matrices are:

Addition: Adding two matrices of the same size by adding corresponding elements.
Subtraction: Subtracting one matrix from another by subtracting corresponding elements.
Scalar multiplication: Multiplying a matrix by a scalar by multiplying each element of the matrix by the scalar.
Matrix multiplication: Multiplying two matrices by multiplying corresponding elements and summing the products.

Q: How do I perform matrix addition?

A: To perform matrix addition, you need to add corresponding elements of the two matrices. For example, if you have two matrices:

{ \left[\begin{array}{cc} 2 & 3 \\ 4 & 5 \end{array}\right] + \left[\begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}\right] \}

You would add the corresponding elements to get:

{ \left[\begin{array}{cc} 2 + 1 & 3 + 2 \\ 4 + 3 & 5 + 4 \end{array}\right] \}

$= \left[\begin{array}{cc} 3 & 5 \\ 7 & 9 \end{array}\right]$

Q: How do I perform matrix subtraction?

A: To perform matrix subtraction, you need to subtract corresponding elements of the two matrices. For example, if you have two matrices:

{ \left[\begin{array}{cc} 2 & 3 \\ 4 & 5 \end{array}\right] - \left[\begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}\right] \}

You would subtract the corresponding elements to get:

{ \left[\begin{array}{cc} 2 - 1 & 3 - 2 \\ 4 - 3 & 5 - 4 \end{array}\right] \}

$= \left[\begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array}\right]$

Q: How do I perform scalar multiplication?

A: To perform scalar multiplication, you need to multiply each element of the matrix by the scalar. For example, if you have a matrix:

{ \left[\begin{array}{cc} 2 & 3 \\ 4 & 5 \end{array}\right] \}

And you want to multiply it by 2, you would multiply each element by 2 to get:

{ \left[\begin{array}{cc} 2(2) & 2(3) \\ 2(4) & 2(5) \end{array}\right] \}

$= \left[\begin{array}{cc} 4 & 6 \\ 8 & 10 \end{array}\right]$

Q: How do I perform matrix multiplication?

A: To perform matrix multiplication, you need to multiply corresponding elements of the two matrices and sum the products. For example, if you have two matrices:

{ \left[\begin{array}{cc} 2 & 3 \\ 4 & 5 \end{array}\right] \left[\begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}\right] \}

You would multiply the corresponding elements and sum the products to get:

{ \left[\begin{array}{cc} 2(1) + 3(3) & 2(2) + 3(4) \\ 4(1) + 5(3) & 4(2) + 5(4) \end{array}\right] \}

$= \left[\begin{array}{cc} 2 + 9 & 4 + 12 \\ 4 + 15 & 8 + 20 \end{array}\right]$

$= \left[\begin{array}{cc} 11 & 16 \\ 19 & 28 \end{array}\right]$

Conclusion

In conclusion, matrix operations are a fundamental concept in linear algebra, and they have numerous applications in various fields. By understanding the basic operations on matrices, you can perform matrix addition, subtraction, scalar multiplication, and matrix multiplication.