Determine What Type Of Matrix Mateus Created From The Given Elements:${ \begin{array}{ccc} a_{11} = 32 & A_{12} = 10 & A_{13} = -9 \ a_{21} = 5 & A_{22} = 7.5 & A_{23} = 4 \ a_{31} = -0.5 & A_{32} = 6 & A_{33} = 16 \ \end{array} }$A. A

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Introduction

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrices are used to represent systems of linear equations, and they have numerous applications in various fields, including physics, engineering, and computer science. In this article, we will determine the type of matrix created by Mateus from the given elements.

The Given Matrix

The given matrix is:

{ \begin{array}{ccc} a_{11} = 32 & a_{12} = 10 & a_{13} = -9 \\ a_{21} = 5 & a_{22} = 7.5 & a_{23} = 4 \\ a_{31} = -0.5 & a_{32} = 6 & a_{33} = 16 \\ \end{array} \}

Types of Matrices

There are several types of matrices, including:

  • Square Matrix: A square matrix is a matrix with the same number of rows and columns. In other words, it is a matrix where the number of rows is equal to the number of columns.
  • Rectangular Matrix: A rectangular matrix is a matrix with a different number of rows and columns. In other words, it is a matrix where the number of rows is not equal to the number of columns.
  • Diagonal Matrix: A diagonal matrix is a square matrix where all the elements outside the main diagonal are zero.
  • Identity Matrix: An identity matrix is a square matrix where all the elements on the main diagonal are ones, and all the elements outside the main diagonal are zeros.
  • Symmetric Matrix: A symmetric matrix is a square matrix where the elements on one side of the main diagonal are the same as the elements on the other side of the main diagonal.
  • Skew-Symmetric Matrix: A skew-symmetric matrix is a square matrix where the elements on one side of the main diagonal are the negative of the elements on the other side of the main diagonal.

Determining the Type of Matrix

To determine the type of matrix created by Mateus, we need to examine the given matrix.

  • Number of Rows and Columns: The given matrix has 3 rows and 3 columns, which means it is a square matrix.
  • Elements on the Main Diagonal: The elements on the main diagonal are a11=32a_{11} = 32, a22=7.5a_{22} = 7.5, and a33=16a_{33} = 16. Since these elements are not all ones, the matrix is not an identity matrix.
  • Elements Outside the Main Diagonal: The elements outside the main diagonal are a12=10a_{12} = 10, a13=−9a_{13} = -9, a21=5a_{21} = 5, a23=4a_{23} = 4, a31=−0.5a_{31} = -0.5, and a32=6a_{32} = 6. Since these elements are not all zeros, the matrix is not a diagonal matrix.
  • Symmetry: The matrix is not symmetric because the elements on one side of the main diagonal are not the same as the elements on the other side of the main diagonal.
  • Skew-Symmetry: The matrix is not skew-symmetric because the elements on one side of the main diagonal are not the negative of the elements on the other side of the main diagonal.

Conclusion

Based on the given elements, we can conclude that the matrix created by Mateus is a square matrix. However, it is not a diagonal matrix, identity matrix, symmetric matrix, or skew-symmetric matrix.

Properties of the Matrix

The given matrix has several properties, including:

  • Determinant: The determinant of the matrix is calculated as follows:

{ \begin{vmatrix} 32 & 10 & -9 \\ 5 & 7.5 & 4 \\ -0.5 & 6 & 16 \\ \end{vmatrix} \}

Using the formula for the determinant of a 3x3 matrix, we get:

{ \begin{align*} \det(A) &= 32(7.5 \cdot 16 - 4 \cdot 6) - 10(5 \cdot 16 - 4 \cdot (-0.5)) + (-9)(5 \cdot 6 - 7.5 \cdot (-0.5)) \\ &= 32(120 - 24) - 10(80 + 2) - 9(30 + 3.75) \\ &= 32 \cdot 96 - 10 \cdot 82 - 9 \cdot 33.75 \\ &= 3072 - 820 - 302.25 \\ &= 1949.75 \end{align*} \}

The determinant of the matrix is 1949.75.

  • Inverse Matrix: The inverse of the matrix is calculated as follows:

{ A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} 7.5 \cdot 16 - 4 \cdot 6 & -(5 \cdot 16 - 4 \cdot (-0.5)) & 5 \cdot 6 - 7.5 \cdot (-0.5) \\ -(10 \cdot 16 - (-9) \cdot 6) & 32 \cdot 16 - (-9) \cdot (-0.5) & -(32 \cdot 6 - 10 \cdot (-0.5)) \\ (10 \cdot 4 - 7.5 \cdot 5) & -(32 \cdot 4 - 7.5 \cdot 5) & 32 \cdot 7.5 - 10 \cdot 5 \end{bmatrix} \}

Using the formula for the inverse of a 3x3 matrix, we get:

{ A^{-1} = \frac{1}{1949.75} \begin{bmatrix} 120 - 24 & -(80 + 2) & 30 + 3.75 \\ -(160 + 54) & 512 + 4.5 & -(512 - 5) \\ (40 - 37.5) & -(128 - 37.5) & 240 - 50 \end{bmatrix} \}

The inverse of the matrix is:

{ A^{-1} = \begin{bmatrix} 0.096 & -0.082 & 0.033 \\ -0.214 & 0.264 & 0.206 \\ 0.0025 & -0.0905 & 0.123 \end{bmatrix} \}

Conclusion

In conclusion, the matrix created by Mateus is a square matrix with a determinant of 1949.75 and an inverse matrix of:

{ A^{-1} = \begin{bmatrix} 0.096 & -0.082 & 0.033 \\ -0.214 & 0.264 & 0.206 \\ 0.0025 & -0.0905 & 0.123 \end{bmatrix} \}

References

  • [1] "Matrix Algebra" by David C. Lay
  • [2] "Linear Algebra and Its Applications" by Gilbert Strang
  • [3] "Matrix Theory" by Richard Bellman

Appendix

The given matrix is:

{ \begin{array}{ccc} a_{11} = 32 & a_{12} = 10 & a_{13} = -9 \\ a_{21} = 5 & a_{22} = 7.5 & a_{23} = 4 \\ a_{31} = -0.5 & a_{32} = 6 & a_{33} = 16 \\ \end{array} \}

The determinant of the matrix is 1949.75, and the inverse of the matrix is:

{ A^{-1} = \begin{bmatrix} 0.096 & -0.082 & 0.033 \\ -0.214 & 0.264 & 0.206 \\ 0.0025 & -0.0905 & 0.123 \end{bmatrix} \}

Q: What type of matrix is the one created by Mateus?

A: The matrix created by Mateus is a square matrix, meaning it has the same number of rows and columns.

Q: What are the dimensions of the matrix?

A: The matrix has 3 rows and 3 columns.

Q: What is the determinant of the matrix?

A: The determinant of the matrix is 1949.75.

Q: What is the inverse of the matrix?

A: The inverse of the matrix is:

{ A^{-1} = \begin{bmatrix} 0.096 & -0.082 & 0.033 \\ -0.214 & 0.264 & 0.206 \\ 0.0025 & -0.0905 & 0.123 \end{bmatrix} \}

Q: Is the matrix symmetric?

A: No, the matrix is not symmetric because the elements on one side of the main diagonal are not the same as the elements on the other side of the main diagonal.

Q: Is the matrix skew-symmetric?

A: No, the matrix is not skew-symmetric because the elements on one side of the main diagonal are not the negative of the elements on the other side of the main diagonal.

Q: Can you provide more information about the properties of the matrix?

A: Yes, the matrix has several properties, including:

  • Determinant: The determinant of the matrix is 1949.75.
  • Inverse Matrix: The inverse of the matrix is:

{ A^{-1} = \begin{bmatrix} 0.096 & -0.082 & 0.033 \\ -0.214 & 0.264 & 0.206 \\ 0.0025 & -0.0905 & 0.123 \end{bmatrix} \}

Q: How can I use the matrix in real-world applications?

A: The matrix can be used in various real-world applications, such as:

  • Linear Algebra: The matrix can be used to solve systems of linear equations.
  • Computer Science: The matrix can be used in computer graphics, machine learning, and data analysis.
  • Physics: The matrix can be used to describe the motion of objects in space.

Q: Can you provide more information about the matrix theory?

A: Yes, the matrix theory is a branch of mathematics that deals with the study of matrices. Matrices are used to represent systems of linear equations, and they have numerous applications in various fields, including physics, engineering, and computer science.

Q: What are some common types of matrices?

A: Some common types of matrices include:

  • Square Matrix: A square matrix is a matrix with the same number of rows and columns.
  • Rectangular Matrix: A rectangular matrix is a matrix with a different number of rows and columns.
  • Diagonal Matrix: A diagonal matrix is a square matrix where all the elements outside the main diagonal are zero.
  • Identity Matrix: An identity matrix is a square matrix where all the elements on the main diagonal are ones, and all the elements outside the main diagonal are zeros.
  • Symmetric Matrix: A symmetric matrix is a square matrix where the elements on one side of the main diagonal are the same as the elements on the other side of the main diagonal.
  • Skew-Symmetric Matrix: A skew-symmetric matrix is a square matrix where the elements on one side of the main diagonal are the negative of the elements on the other side of the main diagonal.

Q: Can you provide more information about the matrix operations?

A: Yes, the matrix operations include:

  • Matrix Addition: Matrix addition is the process of adding two or more matrices.
  • Matrix Multiplication: Matrix multiplication is the process of multiplying two or more matrices.
  • Matrix Transpose: Matrix transpose is the process of swapping the rows and columns of a matrix.
  • Matrix Inverse: Matrix inverse is the process of finding the inverse of a matrix.

Q: Can you provide more information about the matrix applications?

A: Yes, the matrix applications include:

  • Linear Algebra: Matrices are used to solve systems of linear equations.
  • Computer Science: Matrices are used in computer graphics, machine learning, and data analysis.
  • Physics: Matrices are used to describe the motion of objects in space.
  • Engineering: Matrices are used to design and analyze systems, such as electrical circuits and mechanical systems.

Conclusion

In conclusion, the matrix created by Mateus is a square matrix with a determinant of 1949.75 and an inverse matrix of:

{ A^{-1} = \begin{bmatrix} 0.096 & -0.082 & 0.033 \\ -0.214 & 0.264 & 0.206 \\ 0.0025 & -0.0905 & 0.123 \end{bmatrix} \}

The matrix has several properties, including the determinant and inverse matrix. The matrix can be used in various real-world applications, such as linear algebra, computer science, physics, and engineering.