Which Matrix Multiplication Is Possible?A. $\left[\begin{array}{ll}0 & 3\end{array}\right] \times \left[\begin{array}{ll}1 & -4\end{array}\right]B. $\left[\begin{array}{c}3 \ -2\end{array}\right] \times \left[\begin{array}{cc}-1 & 0 \ 0

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Introduction

Matrix multiplication is a fundamental operation in linear algebra, used to combine two matrices into a new matrix. However, not all matrix combinations are possible, and understanding the rules governing matrix multiplication is crucial for accurate calculations. In this article, we will explore the possibilities of matrix multiplication, focusing on the given options A and B.

Matrix Multiplication Rules

Before diving into the possibilities of matrix multiplication, it's essential to understand the rules governing this operation. Matrix multiplication is only possible when the number of columns in the first matrix matches the number of rows in the second matrix. This is known as the "inner dimension" rule.

**Option A: [03]×[1−4]\left[\begin{array}{ll}0 & 3\end{array}\right] \times \left[\begin{array}{ll}1 & -4\end{array}\right]

Let's examine option A, where we have a 1x2 matrix multiplied by a 1x2 matrix.

  • The first matrix has 1 row and 2 columns.
  • The second matrix has 1 row and 2 columns.

Since the number of columns in the first matrix (2) matches the number of rows in the second matrix (2), matrix multiplication is possible. The resulting matrix will have the same number of rows as the first matrix (1) and the same number of columns as the second matrix (2).

Performing Matrix Multiplication for Option A

To perform matrix multiplication for option A, we multiply the corresponding elements of the two matrices.

[03]×[1−4]=[0×1+3×−40×−4+3×1]\left[\begin{array}{ll}0 & 3\end{array}\right] \times \left[\begin{array}{ll}1 & -4\end{array}\right] = \left[\begin{array}{ll}0 \times 1 + 3 \times -4 & 0 \times -4 + 3 \times 1\end{array}\right]

=[−123]= \left[\begin{array}{ll}-12 & 3\end{array}\right]

**Option B: [3−2]×[−1001]\left[\begin{array}{c}3 \\ -2\end{array}\right] \times \left[\begin{array}{cc}-1 & 0 \\ 0 & 1\end{array}\right]

Now, let's examine option B, where we have a 2x1 matrix multiplied by a 2x2 matrix.

  • The first matrix has 2 rows and 1 column.
  • The second matrix has 2 rows and 2 columns.

Since the number of columns in the first matrix (1) does not match the number of rows in the second matrix (2), matrix multiplication is not possible.

Conclusion

In conclusion, matrix multiplication is only possible when the number of columns in the first matrix matches the number of rows in the second matrix. In option A, we have a 1x2 matrix multiplied by a 1x2 matrix, making matrix multiplication possible. The resulting matrix has 1 row and 2 columns. In option B, we have a 2x1 matrix multiplied by a 2x2 matrix, making matrix multiplication not possible.

Matrix Multiplication: Key Takeaways

  • Matrix multiplication is only possible when the number of columns in the first matrix matches the number of rows in the second matrix.
  • The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.
  • Matrix multiplication is a fundamental operation in linear algebra, used to combine two matrices into a new matrix.

Frequently Asked Questions

Q: What are the rules governing matrix multiplication?

A: Matrix multiplication is only possible when the number of columns in the first matrix matches the number of rows in the second matrix.

Q: What is the resulting matrix in option A?

A: The resulting matrix in option A is a 1x2 matrix with elements [-12, 3].

Q: Is matrix multiplication possible in option B?

A: No, matrix multiplication is not possible in option B because the number of columns in the first matrix (1) does not match the number of rows in the second matrix (2).

Q: What is the importance of matrix multiplication in linear algebra?

Introduction

Matrix multiplication is a fundamental operation in linear algebra, used to combine two matrices into a new matrix. However, understanding the rules governing matrix multiplication can be challenging, especially for beginners. In this article, we will provide a comprehensive Q&A guide to help you understand matrix multiplication and its applications.

Q: What is matrix multiplication?

A: Matrix multiplication is a mathematical operation that combines two matrices into a new matrix. It is a fundamental operation in linear algebra, used to solve systems of linear equations, find the inverse of a matrix, and perform other linear algebra operations.

Q: What are the rules governing matrix multiplication?

A: Matrix multiplication is only possible when the number of columns in the first matrix matches the number of rows in the second matrix. This is known as the "inner dimension" rule.

Q: What is the resulting matrix in matrix multiplication?

A: The resulting matrix in matrix multiplication has the same number of rows as the first matrix and the same number of columns as the second matrix.

Q: Can matrix multiplication be performed with matrices of different dimensions?

A: No, matrix multiplication can only be performed with matrices of compatible dimensions. The number of columns in the first matrix must match the number of rows in the second matrix.

Q: What is the difference between matrix multiplication and matrix addition?

A: Matrix multiplication combines two matrices into a new matrix by multiplying corresponding elements, while matrix addition combines two matrices into a new matrix by adding corresponding elements.

Q: Can matrix multiplication be used to solve systems of linear equations?

A: Yes, matrix multiplication can be used to solve systems of linear equations. By multiplying the coefficient matrix by the variable matrix, we can obtain the product matrix, which represents the system of linear equations.

Q: What is the importance of matrix multiplication in linear algebra?

A: Matrix multiplication is a fundamental operation in linear algebra, used to solve systems of linear equations, find the inverse of a matrix, and perform other linear algebra operations.

Q: Can matrix multiplication be used to find the inverse of a matrix?

A: Yes, matrix multiplication can be used to find the inverse of a matrix. By multiplying the coefficient matrix by the inverse matrix, we can obtain the identity matrix.

Q: What are some common applications of matrix multiplication?

A: Some common applications of matrix multiplication include:

  • Solving systems of linear equations
  • Finding the inverse of a matrix
  • Performing linear transformations
  • Solving differential equations

Q: Can matrix multiplication be used in machine learning?

A: Yes, matrix multiplication is used extensively in machine learning, particularly in deep learning. It is used to perform linear transformations, weight updates, and other operations.

Q: What are some common mistakes to avoid when performing matrix multiplication?

A: Some common mistakes to avoid when performing matrix multiplication include:

  • Not checking the dimensions of the matrices before performing multiplication
  • Not using the correct order of multiplication
  • Not handling matrix multiplication errors properly

Conclusion

In conclusion, matrix multiplication is a fundamental operation in linear algebra, used to combine two matrices into a new matrix. Understanding the rules governing matrix multiplication is essential for accurate calculations and applications. By following the guidelines and best practices outlined in this article, you can ensure that your matrix multiplication operations are accurate and efficient.

Matrix Multiplication: Key Takeaways

  • Matrix multiplication is only possible when the number of columns in the first matrix matches the number of rows in the second matrix.
  • The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.
  • Matrix multiplication is a fundamental operation in linear algebra, used to solve systems of linear equations, find the inverse of a matrix, and perform other linear algebra operations.

Frequently Asked Questions

Q: What is matrix multiplication?

A: Matrix multiplication is a mathematical operation that combines two matrices into a new matrix.

Q: What are the rules governing matrix multiplication?

A: Matrix multiplication is only possible when the number of columns in the first matrix matches the number of rows in the second matrix.

Q: Can matrix multiplication be used to solve systems of linear equations?

A: Yes, matrix multiplication can be used to solve systems of linear equations.

Q: What is the importance of matrix multiplication in linear algebra?

A: Matrix multiplication is a fundamental operation in linear algebra, used to solve systems of linear equations, find the inverse of a matrix, and perform other linear algebra operations.

Q: Can matrix multiplication be used in machine learning?

A: Yes, matrix multiplication is used extensively in machine learning, particularly in deep learning.

Q: What are some common mistakes to avoid when performing matrix multiplication?

A: Some common mistakes to avoid when performing matrix multiplication include not checking the dimensions of the matrices before performing multiplication, not using the correct order of multiplication, and not handling matrix multiplication errors properly.