Calculate The Matrix Product:$\[ \left[\begin{array}{cc}2 & -1 \\ -6 & 1\end{array}\right] \cdot \left[\begin{array}{cc}4 & 4 \\ -3 & -5\end{array}\right] \\]

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Introduction

In linear algebra, matrix multiplication is a fundamental operation used to combine two matrices to form a new matrix. The resulting matrix, also known as the product matrix, contains the dot products of the rows of the first matrix with the columns of the second matrix. In this article, we will calculate the matrix product of two given matrices using the standard matrix multiplication algorithm.

Matrix Multiplication Algorithm

The matrix multiplication algorithm involves the following steps:

  1. Check if the matrices can be multiplied: The number of columns in the first matrix must be equal to the number of rows in the second matrix.
  2. Create a new matrix with the correct dimensions: The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.
  3. Calculate the dot products: For each element in the resulting matrix, calculate the dot product of the corresponding row in the first matrix and the corresponding column in the second matrix.

Calculating the Matrix Product

Let's calculate the matrix product of the two given matrices:

{ \left[\begin{array}{cc}2 & -1 \\ -6 & 1\end{array}\right] \cdot \left[\begin{array}{cc}4 & 4 \\ -3 & -5\end{array}\right] \}$ $ ### Step 1: Check if the matrices can be multiplied The first matrix has 2 columns and the second matrix has 2 rows, so they can be multiplied. ### Step 2: Create a new matrix with the correct dimensions The resulting matrix will have 2 rows and 2 columns. ### Step 3: Calculate the dot products To calculate the dot product of the first row of the first matrix and the first column of the second matrix, we multiply the corresponding elements and add them together: (2)(4) + (-1)(-3) = 8 + 3 = 11 To calculate the dot product of the first row of the first matrix and the second column of the second matrix, we multiply the corresponding elements and add them together: (2)(4) + (-1)(-5) = 8 + 5 = 13 To calculate the dot product of the second row of the first matrix and the first column of the second matrix, we multiply the corresponding elements and add them together: (-6)(4) + (1)(-3) = -24 - 3 = -27 To calculate the dot product of the second row of the first matrix and the second column of the second matrix, we multiply the corresponding elements and add them together: (-6)(4) + (1)(-5) = -24 - 5 = -29 ### Step 4: Create the resulting matrix The resulting matrix is: ${ \left[\begin{array}{cc}11 & 13 \\ -27 & -29\end{array}\right] \}$ $ **Conclusion** ---------- In this article, we calculated the matrix product of two given matrices using the standard matrix multiplication algorithm. We checked if the matrices can be multiplied, created a new matrix with the correct dimensions, and calculated the dot products of the rows of the first matrix with the columns of the second matrix. The resulting matrix contains the dot products of the rows of the first matrix with the columns of the second matrix. **Matrix Multiplication Formula** --------------------------- The matrix multiplication formula is: ${ \left[\begin{array}{cc}a & b \\ c & d\end{array}\right] \cdot \left[\begin{array}{cc}e & f \\ g & h\end{array}\right] \}$ = \left[\begin{array}{cc}ae + bg & af + bh \\ ce + dg & cf + dh\end{array}\right] \\] $ **Example Use Cases** -------------------- Matrix multiplication has many practical applications in various fields, including: * **Linear Algebra**: Matrix multiplication is used to solve systems of linear equations, find the inverse of a matrix, and calculate the determinant of a matrix. * **Computer Graphics**: Matrix multiplication is used to perform transformations on 2D and 3D objects, such as rotations, translations, and scaling. * **Machine Learning**: Matrix multiplication is used in neural networks to perform forward and backward passes, and to update the model parameters during training. **Common Mistakes** ------------------ When performing matrix multiplication, it's easy to make mistakes. Here are some common mistakes to avoid: * **Incorrect dimensions**: Make sure the number of columns in the first matrix is equal to the number of rows in the second matrix. * **Incorrect calculations**: Double-check your calculations to ensure you're getting the correct dot products. * **Incorrect matrix dimensions**: Make sure the resulting matrix has the correct dimensions. **Conclusion** ---------- In conclusion, matrix multiplication is a fundamental operation in linear algebra that has many practical applications in various fields. By following the standard matrix multiplication algorithm and avoiding common mistakes, you can perform matrix multiplication accurately and efficiently.<br/> **Matrix Multiplication Q&A** ========================== **Frequently Asked Questions** --------------------------- ### Q: What is matrix multiplication? A: Matrix multiplication is a fundamental operation in linear algebra that involves combining two matrices to form a new matrix. The resulting matrix, also known as the product matrix, contains the dot products of the rows of the first matrix with the columns of the second matrix. ### Q: What are the conditions for matrix multiplication? A: The number of columns in the first matrix must be equal to the number of rows in the second matrix. This is a necessary condition for matrix multiplication to be possible. ### Q: How do I calculate the dot product of two matrices? A: To calculate the dot product of two matrices, you multiply the corresponding elements of the rows of the first matrix with the corresponding elements of the columns of the second matrix, and then add the results together. ### Q: What is the resulting matrix in matrix multiplication? A: The resulting matrix in matrix multiplication is a new matrix that contains the dot products of the rows of the first matrix with the columns of the second matrix. ### Q: Can I multiply two matrices of different dimensions? A: No, you cannot multiply two matrices of different dimensions. The number of columns in the first matrix must be equal to the number of rows in the second matrix. ### Q: What is the difference between matrix multiplication and matrix addition? A: Matrix multiplication involves combining two matrices to form a new matrix by taking the dot products of the rows of the first matrix with the columns of the second matrix. Matrix addition involves adding corresponding elements of two matrices to form a new matrix. ### Q: What are some common applications of matrix multiplication? A: Matrix multiplication has many practical applications in various fields, including linear algebra, computer graphics, and machine learning. ### Q: How do I perform matrix multiplication in a programming language? A: The syntax for matrix multiplication in a programming language will depend on the language and the library or framework being used. However, most programming languages provide a built-in function or method for matrix multiplication. ### Q: What are some common mistakes to avoid when performing matrix multiplication? A: Some common mistakes to avoid when performing matrix multiplication include incorrect dimensions, incorrect calculations, and incorrect matrix dimensions. ### Q: Can I multiply a matrix by a scalar? A: Yes, you can multiply a matrix by a scalar. This is known as scalar multiplication. ### Q: What is the result of multiplying a matrix by a scalar? A: The result of multiplying a matrix by a scalar is a new matrix where each element of the original matrix is multiplied by the scalar. ### Q: Can I multiply two scalars? A: Yes, you can multiply two scalars. This is a basic arithmetic operation. ### Q: What is the result of multiplying two scalars? A: The result of multiplying two scalars is a new scalar that is the product of the two original scalars. **Conclusion** ---------- In conclusion, matrix multiplication is a fundamental operation in linear algebra that has many practical applications in various fields. By understanding the conditions for matrix multiplication, how to calculate the dot product, and common applications, you can perform matrix multiplication accurately and efficiently. **Matrix Multiplication Resources** ------------------------------ * **Linear Algebra Textbooks**: There are many excellent linear algebra textbooks that cover matrix multiplication in detail. * **Online Courses**: There are many online courses that cover matrix multiplication, including those on Coursera, edX, and Udemy. * **Programming Libraries**: Many programming libraries, such as NumPy and pandas, provide built-in functions for matrix multiplication. * **Mathematical Software**: Mathematical software, such as MATLAB and Mathematica, provide built-in functions for matrix multiplication. **Matrix Multiplication Practice Problems** ----------------------------------------- * **Matrix Multiplication Examples**: Practice multiplying matrices of different dimensions to get a feel for the operation. * **Matrix Multiplication Exercises**: Practice solving matrix multiplication problems to improve your skills. * **Matrix Multiplication Projects**: Work on projects that involve matrix multiplication, such as image processing or data analysis. **Conclusion** ---------- In conclusion, matrix multiplication is a fundamental operation in linear algebra that has many practical applications in various fields. By understanding the conditions for matrix multiplication, how to calculate the dot product, and common applications, you can perform matrix multiplication accurately and efficiently.