Consider The Equation:${ \left[\begin{array}{cc} 7 & 4 \ -3 & 2 \end{array}\right] + \left[\begin{array}{ll} w & X \ y & Z \end{array}\right] = \left[\begin{array}{cc} 7 & 4 \ -3 & 2 \end{array}\right] }$Which Statement Is True?A. The

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Introduction

In the realm of mathematics, matrices are a fundamental concept used to represent systems of linear equations, linear transformations, and more. Matrix addition is a crucial operation that allows us to combine two or more matrices to obtain a new matrix. In this article, we will delve into the world of matrix addition, exploring the rules and properties that govern this operation. We will examine a specific equation involving matrix addition and determine which statement is true.

Matrix Addition Rules

Matrix addition is a straightforward operation that involves adding corresponding elements of two matrices. Given two matrices A and B, the sum of A and B, denoted as A + B, is a new matrix C, where each element of C is the sum of the corresponding elements of A and B.

Notation

Let's consider two matrices A and B, where:

A = a11a12a21a22{\begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}}

B = b11b12b21b22{\begin{array}{cc} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array}}

The sum of A and B, denoted as A + B, is:

A + B = a11+b11a12+b12a21+b21a22+b22{\begin{array}{cc} a_{11} + b_{11} & a_{12} + b_{12} \\ a_{21} + b_{21} & a_{22} + b_{22} \end{array}}

Properties of Matrix Addition

Matrix addition has several important properties that govern this operation. These properties include:

  • Commutativity: The order of matrix addition does not affect the result. In other words, A + B = B + A.
  • Associativity: The order in which we add matrices does not affect the result. In other words, (A + B) + C = A + (B + C).
  • Distributivity: Matrix addition distributes over scalar multiplication. In other words, c(A + B) = cA + cB, where c is a scalar.

The Equation

Now, let's consider the equation:

74−32{\begin{array}{cc} 7 & 4 \\ -3 & 2 \end{array}} + wxyz{\begin{array}{ll} w & x \\ y & z \end{array}} = 74−32{\begin{array}{cc} 7 & 4 \\ -3 & 2 \end{array}}

To determine which statement is true, we need to examine the properties of matrix addition.

Analysis

Let's analyze the equation:

74−32{\begin{array}{cc} 7 & 4 \\ -3 & 2 \end{array}} + wxyz{\begin{array}{ll} w & x \\ y & z \end{array}} = 74−32{\begin{array}{cc} 7 & 4 \\ -3 & 2 \end{array}}

For the equation to be true, the sum of the two matrices on the left-hand side must be equal to the matrix on the right-hand side. This means that the elements of the two matrices on the left-hand side must be equal to the corresponding elements of the matrix on the right-hand side.

Conclusion

Based on the analysis, we can conclude that the statement "The sum of the two matrices on the left-hand side is equal to the matrix on the right-hand side" is true. This is because the elements of the two matrices on the left-hand side must be equal to the corresponding elements of the matrix on the right-hand side.

Final Answer

The final answer is that the statement "The sum of the two matrices on the left-hand side is equal to the matrix on the right-hand side" is true.

Matrix Addition Rules

Matrix addition is a straightforward operation that involves adding corresponding elements of two matrices. Given two matrices A and B, the sum of A and B, denoted as A + B, is a new matrix C, where each element of C is the sum of the corresponding elements of A and B.

Properties of Matrix Addition

Matrix addition has several important properties that govern this operation. These properties include:

  • Commutativity: The order of matrix addition does not affect the result. In other words, A + B = B + A.
  • Associativity: The order in which we add matrices does not affect the result. In other words, (A + B) + C = A + (B + C).
  • Distributivity: Matrix addition distributes over scalar multiplication. In other words, c(A + B) = cA + cB, where c is a scalar.

Matrix Addition Rules

Matrix addition is a straightforward operation that involves adding corresponding elements of two matrices. Given two matrices A and B, the sum of A and B, denoted as A + B, is a new matrix C, where each element of C is the sum of the corresponding elements of A and B.

Properties of Matrix Addition

Matrix addition has several important properties that govern this operation. These properties include:

  • Commutativity: The order of matrix addition does not affect the result. In other words, A + B = B + A.
  • Associativity: The order in which we add matrices does not affect the result. In other words, (A + B) + C = A + (B + C).
  • Distributivity: Matrix addition distributes over scalar multiplication. In other words, c(A + B) = cA + cB, where c is a scalar.

Matrix Addition Rules

Matrix addition is a straightforward operation that involves adding corresponding elements of two matrices. Given two matrices A and B, the sum of A and B, denoted as A + B, is a new matrix C, where each element of C is the sum of the corresponding elements of A and B.

Properties of Matrix Addition

Matrix addition has several important properties that govern this operation. These properties include:

  • Commutativity: The order of matrix addition does not affect the result. In other words, A + B = B + A.
  • Associativity: The order in which we add matrices does not affect the result. In other words, (A + B) + C = A + (B + C).
  • Distributivity: Matrix addition distributes over scalar multiplication. In other words, c(A + B) = cA + cB, where c is a scalar.
    Matrix Addition: A Mathematical Exploration =====================================================

Q&A: Matrix Addition

Q: What is matrix addition?

A: Matrix addition is a mathematical operation that involves adding corresponding elements of two matrices. Given two matrices A and B, the sum of A and B, denoted as A + B, is a new matrix C, where each element of C is the sum of the corresponding elements of A and B.

Q: What are the rules of matrix addition?

A: The rules of matrix addition are as follows:

  • Commutativity: The order of matrix addition does not affect the result. In other words, A + B = B + A.
  • Associativity: The order in which we add matrices does not affect the result. In other words, (A + B) + C = A + (B + C).
  • Distributivity: Matrix addition distributes over scalar multiplication. In other words, c(A + B) = cA + cB, where c is a scalar.

Q: What are the properties of matrix addition?

A: The properties of matrix addition are as follows:

  • Commutativity: The order of matrix addition does not affect the result. In other words, A + B = B + A.
  • Associativity: The order in which we add matrices does not affect the result. In other words, (A + B) + C = A + (B + C).
  • Distributivity: Matrix addition distributes over scalar multiplication. In other words, c(A + B) = cA + cB, where c is a scalar.

Q: How do I add two matrices?

A: To add two matrices, you need to add corresponding elements of the two matrices. For example, if you have two matrices A and B, where:

A = a11a12a21a22{\begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}}

B = b11b12b21b22{\begin{array}{cc} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array}}

The sum of A and B, denoted as A + B, is:

A + B = a11+b11a12+b12a21+b21a22+b22{\begin{array}{cc} a_{11} + b_{11} & a_{12} + b_{12} \\ a_{21} + b_{21} & a_{22} + b_{22} \end{array}}

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

A: Matrix addition and scalar multiplication are two different mathematical operations. Matrix addition involves adding corresponding elements of two matrices, while scalar multiplication involves multiplying each element of a matrix by a scalar.

Q: Can I add a matrix and a scalar?

A: Yes, you can add a matrix and a scalar. In this case, the scalar is multiplied by each element of the matrix.

Q: What are some common applications of matrix addition?

A: Matrix addition has many applications in mathematics, science, and engineering. Some common applications include:

  • Linear algebra: Matrix addition is used to solve systems of linear equations and to find the inverse of a matrix.
  • Computer graphics: Matrix addition is used to perform transformations on 2D and 3D objects.
  • Physics: Matrix addition is used to describe the motion of objects in space.

Conclusion

Matrix addition is a fundamental concept in mathematics that has many applications in science and engineering. By understanding the rules and properties of matrix addition, you can solve systems of linear equations, perform transformations on 2D and 3D objects, and describe the motion of objects in space.