Suppose $A$ And $B$ Are $2 \times 1$ Matrices. Drag And Drop The Correct Dimensions Of The Matrix $A + B$. 1. $1 \times 1$ 2. $2 \times 1$ 3. $1 \times 2$ 4. $2 \times

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Introduction

In the realm of linear algebra, matrices play a crucial role in representing systems of equations and solving problems in various fields such as physics, engineering, and computer science. When working with matrices, it's essential to understand the rules of matrix addition, which is a fundamental operation in matrix algebra. In this article, we will delve into the concept of matrix addition, explore the properties of matrix addition, and determine the correct dimensions of the resulting matrix when two matrices are added together.

What are Matrices?

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrices can be used to represent systems of linear equations, linear transformations, and other mathematical objects. In this article, we will focus on 2x1 matrices, which are matrices with two rows and one column.

Matrix Addition

Matrix addition is a binary operation that takes two matrices as input and produces another matrix as output. The resulting matrix is obtained by adding corresponding elements of the two input matrices. For example, if we have two 2x1 matrices A and B, the resulting matrix A + B will have the same number of rows as A and B, and the same number of columns as A and B.

Properties of Matrix Addition

Matrix addition has several important properties that are essential to understand:

  • Closure: The sum of two matrices is always a matrix.
  • Commutativity: The order of the matrices being added does not affect the result. In other words, A + B = B + A.
  • Associativity: The order in which we add three or more matrices does not affect the result. In other words, (A + B) + C = A + (B + C).
  • Existence of Additive Identity: There exists a matrix, called the additive identity, such that when added to any matrix, the result is the original matrix. The additive identity is a matrix with all elements equal to zero.

Determining the Dimensions of the Resulting Matrix

Now that we have a good understanding of matrix addition, let's focus on determining the dimensions of the resulting matrix when two matrices are added together. Suppose we have two 2x1 matrices A and B. We want to determine the dimensions of the resulting matrix A + B.

Step 1: Identify the Number of Rows

Since both matrices A and B have two rows, the resulting matrix A + B will also have two rows.

Step 2: Identify the Number of Columns

Since both matrices A and B have one column, the resulting matrix A + B will also have one column.

Conclusion

In conclusion, when two 2x1 matrices A and B are added together, the resulting matrix A + B will have two rows and one column, i.e., it will be a 2x1 matrix. Therefore, the correct dimensions of the matrix A + B are 2x1.

Example

To illustrate this concept, let's consider an example. Suppose we have two 2x1 matrices A and B:

A = [1] [2]

B = [3] [4]

When we add these two matrices together, we get:

A + B = [1 + 3] [2 + 4] = [4] = [6]

As we can see, the resulting matrix A + B has two rows and one column, i.e., it is a 2x1 matrix.

Conclusion

In this article, we have explored the concept of matrix addition, properties of matrix addition, and determined the correct dimensions of the resulting matrix when two matrices are added together. We have shown that when two 2x1 matrices A and B are added together, the resulting matrix A + B will have two rows and one column, i.e., it will be a 2x1 matrix. This understanding is essential in various fields such as physics, engineering, and computer science, where matrices are used to represent systems of equations and solve problems.

Final Answer

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions related to matrix addition.

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

A: Matrix addition and scalar multiplication are two different operations in matrix algebra. Matrix addition involves adding two or more matrices together, while scalar multiplication involves multiplying a matrix by a scalar (a number). For example, if we have two 2x1 matrices A and B, the sum A + B is a 2x1 matrix, while the product 2A is a 2x1 matrix.

Q: Can we add two matrices of different dimensions?

A: No, we cannot add two matrices of different dimensions. The number of rows and columns in the resulting matrix must be the same as the number of rows and columns in the input matrices. For example, we cannot add a 2x1 matrix and a 3x1 matrix.

Q: What is the additive identity in matrix algebra?

A: The additive identity in matrix algebra is a matrix with all elements equal to zero. When we add the additive identity to any matrix, the result is the original matrix. For example, if we have a 2x1 matrix A, the additive identity is a 2x1 matrix with all elements equal to zero, i.e., [0] [0]. When we add A to the additive identity, we get A + [0] [0] = A.

Q: Can we add two matrices that are not square?

A: Yes, we can add two matrices that are not square. For example, we can add a 2x1 matrix and a 2x3 matrix. The resulting matrix will have two rows and three columns.

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

A: Matrix addition and matrix multiplication are two different operations in matrix algebra. Matrix addition involves adding two or more matrices together, while matrix multiplication involves multiplying two matrices together. For example, if we have two 2x1 matrices A and B, the sum A + B is a 2x1 matrix, while the product AB is a 2x2 matrix.

Q: Can we add two matrices that have different data types?

A: No, we cannot add two matrices that have different data types. For example, we cannot add a matrix of integers and a matrix of floating-point numbers.

Q: What is the result of adding two matrices that have the same elements?

A: The result of adding two matrices that have the same elements is a matrix with the same elements. For example, if we have two 2x1 matrices A and B with the same elements, i.e., A = [1] [2] and B = [1] [2], the sum A + B is a 2x1 matrix with the same elements, i.e., [1] [2].

Q: Can we add two matrices that have complex elements?

A: Yes, we can add two matrices that have complex elements. For example, if we have two 2x1 matrices A and B with complex elements, i.e., A = [1 + 2i] [3 - 4i] and B = [1 - 2i] [3 + 4i], the sum A + B is a 2x1 matrix with complex elements, i.e., [2] [6].

Conclusion

In this article, we have addressed some of the most frequently asked questions related to matrix addition. We have discussed the difference between matrix addition and scalar multiplication, the additive identity, and the result of adding two matrices that have the same elements. We have also discussed the result of adding two matrices that have complex elements.