What Is The Product?$\[ \left[\begin{array}{cc} 5 & -2 \\ -6 & 2 \end{array}\right] \times \left[\begin{array}{c} -1 \\ -1 \end{array}\right] \\]A. \[$\left[\begin{array}{c}-7 \\ 8\end{array}\right]\$\]B.
Introduction
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. A vector is a one-dimensional array of numbers, symbols, or expressions. When we multiply a matrix by a vector, we get a new vector as a result. This process is called matrix-vector multiplication. In this article, we will explore the concept of matrix-vector multiplication and how to perform it.
What is Matrix-Vector Multiplication?
Matrix-vector multiplication is a fundamental operation in linear algebra, which is a branch of mathematics that deals with vectors and matrices. When we multiply a matrix by a vector, we get a new vector as a result. The resulting vector is a linear combination of the columns of the matrix, where the coefficients of the linear combination are given by the elements of the vector.
How to Perform Matrix-Vector Multiplication
To perform matrix-vector multiplication, we need to follow these steps:
- Identify the Matrix and the Vector: The first step is to identify the matrix and the vector that we want to multiply. In the given problem, the matrix is and the vector is .
- Multiply the Matrix by the Vector: To multiply the matrix by the vector, we need to multiply each element of the matrix by the corresponding element of the vector. The resulting vector will have the same number of elements as the matrix.
- Add the Products: After multiplying each element of the matrix by the corresponding element of the vector, we need to add the products to get the final result.
Example: Multiplying a Matrix by a Vector
Let's use the given problem as an example. We have the matrix and the vector . To multiply the matrix by the vector, we need to follow the steps mentioned above.
Step 1: Multiply the Matrix by the Vector
| -1 | -1 | |
|---|---|---|
| 5 | -5 | 2 |
| -6 | 6 | -2 |
Step 2: Add the Products
To add the products, we need to add the corresponding elements of the two vectors.
| -5 + 6 | 2 - 2 | |
|---|---|---|
| 1 | 0 |
The Final Result
The final result of multiplying the matrix by the vector is .
Conclusion
In this article, we explored the concept of matrix-vector multiplication and how to perform it. We used a specific example to illustrate the steps involved in multiplying a matrix by a vector. Matrix-vector multiplication is a fundamental operation in linear algebra, and it has many applications in science, engineering, and other fields.
What is the Product of the Given Matrix and Vector?
To find the product of the given matrix and vector, we need to follow the steps mentioned above.
| -1 | -1 | |
|---|---|---|
| 5 | -5 | 2 |
| -6 | 6 | -2 |
Step 1: Multiply the Matrix by the Vector
To multiply the matrix by the vector, we need to multiply each element of the matrix by the corresponding element of the vector.
| -1 | -1 | |
|---|---|---|
| 5 | -5 | 2 |
| -6 | 6 | -2 |
Step 2: Add the Products
To add the products, we need to add the corresponding elements of the two vectors.
| -5 + 6 | 2 - 2 | |
|---|---|---|
| 1 | 0 |
The Final Result
The final result of multiplying the matrix by the vector is .
Answer
The correct answer is .
Discussion
Matrix-vector multiplication is a fundamental operation in linear algebra, and it has many applications in science, engineering, and other fields. The product of a matrix and a vector is a linear combination of the columns of the matrix, where the coefficients of the linear combination are given by the elements of the vector.
References
- [1] Linear Algebra and Its Applications, Gilbert Strang
- [2] Matrix Algebra, James E. Gentle
- [3] Linear Algebra, David C. Lay
Conclusion
Q: What is matrix-vector multiplication?
A: Matrix-vector multiplication is a fundamental operation in linear algebra, which is a branch of mathematics that deals with vectors and matrices. When we multiply a matrix by a vector, we get a new vector as a result.
Q: How do I perform matrix-vector multiplication?
A: To perform matrix-vector multiplication, you need to follow these steps:
- Identify the Matrix and the Vector: The first step is to identify the matrix and the vector that you want to multiply.
- Multiply the Matrix by the Vector: To multiply the matrix by the vector, you need to multiply each element of the matrix by the corresponding element of the vector.
- Add the Products: After multiplying each element of the matrix by the corresponding element of the vector, you need to add the products to get the final result.
Q: What is the difference between matrix-vector multiplication and matrix-matrix multiplication?
A: Matrix-vector multiplication involves multiplying a matrix by a vector, while matrix-matrix multiplication involves multiplying two matrices. The resulting matrix in matrix-matrix multiplication has the same number of rows as the first matrix and the same number of columns as the second matrix.
Q: Can I multiply a vector by a matrix?
A: Yes, you can multiply a vector by a matrix. However, the resulting vector will have the same number of elements as the number of rows in the matrix.
Q: What are some common applications of matrix-vector multiplication?
A: Matrix-vector multiplication has many applications in science, engineering, and other fields, including:
- Linear Transformations: Matrix-vector multiplication is used to represent linear transformations, which are fundamental in many areas of mathematics and science.
- Data Analysis: Matrix-vector multiplication is used in data analysis to perform operations such as filtering, smoothing, and feature extraction.
- Machine Learning: Matrix-vector multiplication is used in machine learning to perform operations such as neural network computations and data preprocessing.
- Computer Graphics: Matrix-vector multiplication is used in computer graphics to perform operations such as transformations and projections.
Q: How do I handle matrix-vector multiplication with complex numbers?
A: When working with complex numbers, you need to follow the same steps as with real numbers. However, you need to be careful with the signs and the order of operations.
Q: Can I use matrix-vector multiplication with non-square matrices?
A: Yes, you can use matrix-vector multiplication with non-square matrices. However, the resulting vector will have the same number of elements as the number of rows in the matrix.
Q: What are some common mistakes to avoid when performing matrix-vector multiplication?
A: Some common mistakes to avoid when performing matrix-vector multiplication include:
- Incorrect Order of Operations: Make sure to follow the correct order of operations when multiplying the matrix by the vector.
- Incorrect Signs: Be careful with the signs when multiplying the matrix by the vector.
- Incorrect Dimensions: Make sure that the dimensions of the matrix and the vector are compatible for multiplication.
Conclusion
In this article, we answered some frequently asked questions about matrix-vector multiplication. We covered topics such as the definition of matrix-vector multiplication, how to perform it, and some common applications. We also discussed some common mistakes to avoid when performing matrix-vector multiplication.