Select The Correct Answer.$[ A = 2\left[\begin{array}{ccc} -6 & 7 & -4 \ 4 & -4 & 3 \ -2 & 4 & 0 \ 7 & 0 & -2 \end{array}\right], \quad B = -3\left[\begin{array}{ccc} 5 & -9 & 1 \ 3 & 6 & -3 \ 8 & 9 & 1 \ 4 & 9 & -9 \end{array}\right]

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Introduction

In mathematics, matrices are a fundamental concept used to represent linear transformations and solve systems of equations. Matrix operations are essential in various fields, including physics, engineering, and computer science. In this article, we will explore matrix operations and help you select the correct answer for a given problem.

Matrix Addition and Scalar Multiplication

Matrix addition and scalar multiplication are two basic operations used to manipulate matrices. Matrix addition involves adding corresponding elements of two matrices, while scalar multiplication involves multiplying each element of a matrix by a scalar.

Matrix Addition

Matrix addition is a straightforward operation that involves adding corresponding elements of two matrices. The resulting matrix has the same dimensions as the original matrices. For example, consider two matrices A and B with the following elements:

A = 2\left[\begin{array}{ccc} -6 & 7 & -4 \ 4 & -4 & 3 \ -2 & 4 & 0 \ 7 & 0 & -2 \end{array}\right]

B = -3\left[\begin{array}{ccc} 5 & -9 & 1 \ 3 & 6 & -3 \ 8 & 9 & 1 \ 4 & 9 & -9 \end{array}\right]

To add these matrices, we simply add corresponding elements:

A + B = 2\left[\begin{array}{ccc} -6 & 7 & -4 \ 4 & -4 & 3 \ -2 & 4 & 0 \ 7 & 0 & -2 \end{array}\right] + (-3)\left[\begin{array}{ccc} 5 & -9 & 1 \ 3 & 6 & -3 \ 8 & 9 & 1 \ 4 & 9 & -9 \end{array}\right]

A + B = \left[\begin{array}{ccc} -12-15 & 14+27 & -8-3 \ 8-9 & -8-18 & 3+9 \ -4-24 & 8+27 & 0+3 \ 14-12 & 0-27 & -4+27 \end{array}\right]

A + B = \left[\begin{array}{ccc} -27 & 41 & -11 \ -1 & -26 & 12 \ -28 & 35 & 3 \ 2 & -27 & 23 \end{array}\right]

Scalar Multiplication

Scalar multiplication involves multiplying each element of a matrix by a scalar. For example, consider the matrix A with the following elements:

A = 2\left[\begin{array}{ccc} -6 & 7 & -4 \ 4 & -4 & 3 \ -2 & 4 & 0 \ 7 & 0 & -2 \end{array}\right]

To multiply this matrix by a scalar 3, we simply multiply each element by 3:

3A = 3(2)\left[\begin{array}{ccc} -6 & 7 & -4 \ 4 & -4 & 3 \ -2 & 4 & 0 \ 7 & 0 & -2 \end{array}\right]

3A = 6\left[\begin{array}{ccc} -6 & 7 & -4 \ 4 & -4 & 3 \ -2 & 4 & 0 \ 7 & 0 & -2 \end{array}\right]

3A = \left[\begin{array}{ccc} -36 & 42 & -24 \ 24 & -24 & 18 \ -12 & 24 & 0 \ 42 & 0 & -12 \end{array}\right]

Matrix Multiplication

Matrix multiplication is a more complex operation that involves multiplying corresponding elements of two matrices. 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 Example

Consider two matrices A and B with the following elements:

A = 2\left[\begin{array}{ccc} -6 & 7 & -4 \ 4 & -4 & 3 \ -2 & 4 & 0 \ 7 & 0 & -2 \end{array}\right]

B = -3\left[\begin{array}{ccc} 5 & -9 & 1 \ 3 & 6 & -3 \ 8 & 9 & 1 \ 4 & 9 & -9 \end{array}\right]

To multiply these matrices, we need to follow the rules of matrix multiplication. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.

Let's calculate the elements of the resulting matrix:

C = AB

C = \left[\begin{array}{ccc} -36 & 42 & -24 \ 24 & -24 & 18 \ -12 & 24 & 0 \ 42 & 0 & -12 \end{array}\right] \left[\begin{array}{ccc} 5 & -9 & 1 \ 3 & 6 & -3 \ 8 & 9 & 1 \ 4 & 9 & -9 \end{array}\right]

C = \left[\begin{array}{ccc} (-36)(5)+42(3)+(-24)(8)+42(4) & (-36)(-9)+42(6)+(-24)(9)+42(9) & (-36)(1)+42(-3)+(-24)(1)+42(-9) \ 24(5)+(-24)(3)+18(8)+42(4) & 24(-9)+(-24)(6)+18(9)+42(9) & 24(1)+(-24)(-3)+18(1)+42(-9) \ (-12)(5)+24(3)+0(8)+0(4) & (-12)(-9)+24(6)+0(9)+0(9) & (-12)(1)+24(-3)+0(1)+0(-9) \ 42(5)+0(3)+(-12)(8)+0(4) & 42(-9)+0(6)+(-12)(9)+0(9) & 42(1)+0(-3)+(-12)(1)+0(-9) \end{array}\right]

C = \left[\begin{array}{ccc} -180+126-192+168 & 324+252-216+378 & -36-126-24-378 \ 120-72+144+168 & -216-144+162+378 & 24+72+18-378 \ -60+72+0+0 & 108+144+0+0 & -12-72+0+0 \ 210+0-96+0 & -378+0-108+0 & 42+0-12+0 \end{array}\right]

C = \left[\begin{array}{ccc} -78 & 438 & -564 \ 360 & 180 & -324 \ 12 & 252 & -84 \ 114 & -486 & 30 \end{array}\right]

Conclusion

In this article, we explored matrix operations, including matrix addition, scalar multiplication, and matrix multiplication. We also provided examples of these operations and calculated the resulting matrices. Matrix operations are essential in various fields, including physics, engineering, and computer science. By understanding these operations, you can solve complex problems and make informed decisions.

Final Answer

The final answer to the problem is:

A + B = \left[\begin{array}{ccc} -27 & 41 & -11 \ -1 & -26 & 12 \ -28 & 35 & 3 \ 2 & -27 & 23 \end{array}\right]

3A = \left[\begin{array}{ccc} -36 & 42 & -24 \ 24 & -24 & 18 \ -12 & 24 & 0 \ 42 & 0 & -12 \end{array}\right]

C = AB = \left[\begin{array}{ccc} -78 & 438 & -564 \ 360 & 180 & -324 \ 12 & 252 & -84 \ 114 & -486 & 30 \end{array}\right]

Note: The final answer is a matrix, not a single number.

Introduction

In our previous article, we explored matrix operations, including matrix addition, scalar multiplication, and matrix multiplication. In this article, we will answer some frequently asked questions about matrix operations.

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

A: 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 two matrices if they have different dimensions?

A: No, you cannot add two matrices if they have different dimensions. The matrices must have the same dimensions to be added.

Q: Can I multiply two matrices if they have different dimensions?

A: Yes, you can multiply two matrices if they have compatible 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 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 a matrix by a scalar if the scalar is zero?

A: Yes, you can multiply a matrix by a scalar if the scalar is zero. The result will be a matrix where each element is zero.

Q: What is the result of multiplying two matrices?

A: The result of multiplying two matrices is a new matrix where each element is the dot product of a row from the first matrix and a column from the second matrix.

Q: Can I multiply two matrices if they have different dimensions?

A: Yes, you can multiply two matrices if they have compatible 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 scalar multiplication?

A: Matrix multiplication involves multiplying corresponding elements of two matrices, while scalar multiplication involves multiplying each element of a matrix by a scalar.

Q: Can I add a matrix to a scalar?

A: No, you cannot add a matrix to a scalar. The two operations are different and cannot be combined.

Q: Can I multiply a matrix by a matrix if the matrix is a scalar?

A: No, you cannot multiply a matrix by a matrix if the matrix is a scalar. The two operations are different and cannot be combined.

Q: What is the result of multiplying a matrix by a matrix if the matrix is a scalar?

A: The result will be a matrix where each element is the product of the scalar and the corresponding element in the original matrix.

Q: Can I multiply a matrix by a matrix if the matrix is a vector?

A: No, you cannot multiply a matrix by a matrix if the matrix is a vector. The two operations are different and cannot be combined.

Q: What is the result of multiplying a matrix by a matrix if the matrix is a vector?

A: The result will be a vector where each element is the dot product of the vector and the corresponding column in the original matrix.

Conclusion

In this article, we answered some frequently asked questions about matrix operations. We hope that this article has helped to clarify any confusion and provide a better understanding of matrix operations.

Final Answer

The final answer to the question is:

  • Matrix addition involves adding corresponding elements of two matrices.
  • Scalar multiplication involves multiplying each element of a matrix by a scalar.
  • Matrix multiplication involves multiplying corresponding elements of two matrices.
  • The result of multiplying a matrix by a scalar is a new matrix where each element is multiplied by the scalar.
  • The result of multiplying two matrices is a new matrix where each element is the dot product of a row from the first matrix and a column from the second matrix.

Note: The final answer is a summary of the key points discussed in the article.