Matrix and Determinant - Analytical problem on symmetric and skew-symmetric matrix

• An n x n matrix A is said to be symmetric if A = A^T.
• A symmetric matrix can be written in the form A = [a_ij] where a_ij = a_ji for all i and j.
• An n x n matrix A is said to be skew-symmetric if A = -A^T.
• A skew-symmetric matrix can be written in the form A = [a_ij] where a_ij = -a_ji for all i and j. Example: Let’s consider a symmetric matrix A = [[1, 2], [2, 3]].
• A^T = [[1, 2], [2, 3]] which is equal to A.
• Therefore, A is a symmetric matrix.

Properties of Symmetric and Skew-Symmetric Matrices

For two n x n matrices A and B, if A and B are symmetric and k is any scalar, the following properties hold:

• A ± B is also symmetric.
• kA is also symmetric.
• A^T is also symmetric.
• A · B is not necessarily symmetric.
• A^(-1) is not necessarily symmetric.
• The sum of a symmetric matrix and a skew-symmetric matrix is neither symmetric nor skew-symmetric. Example: Let’s consider A = [[1, 2], [2, 3]] and B = [[4, 5], [5, 6]].
• A + B = [[1+4, 2+5], [2+5, 3+6]] = [[5, 7], [7, 9]] which is symmetric.

Properties of Symmetric and Skew-Symmetric Matrices (continued)

For two n x n matrices A and B, if A and B are skew-symmetric and k is any scalar, the following properties hold:

• A ± B is also skew-symmetric.
• kA is also skew-symmetric.
• A^T is also skew-symmetric.
• A · B is skew-symmetric.
• A^(-1) is not necessarily skew-symmetric.
• The sum of a symmetric matrix and a skew-symmetric matrix is neither symmetric nor skew-symmetric. Example: Let’s consider A = [[0, 2], [-2, 0]] and B = [[0, 3], [-3, 0]].
• A + B = [[0+0, 2+3], [-2-3, 0+0]] = [[0, 5], [-5, 0]] which is neither symmetric nor skew-symmetric.

Solving Analytical Problems

To solve analytical problems involving symmetric and skew-symmetric matrices, follow these steps:

1. Identify the given matrix as symmetric or skew-symmetric.

2. Apply the appropriate properties and operations to obtain the desired result.

3. Verify the solutions using the given conditions or by performing necessary calculations. Example: Solve the system of linear equations using matrices:

• x + 2y = 4
• 2x + 3y = 7

Solving Analytical Problems (continued)

1. Write the system of linear equations in matrix form:
   • [1 2] [x] = [4] [2 3] [y] [7]

2. Identify the coefficient matrix as symmetric or skew-symmetric.
   • The coefficient matrix is not symmetric or skew-symmetric.

3. Solve the system of equations using matrix operations:

   • [1 2] [x] = [4] (1) [2 3] [y] [7] (2)

   • Apply Gaussian elimination or matrix inversion method to obtain the solutions.

Gaussian Elimination Method

The Gaussian elimination method for solving systems of linear equations involves the following steps:

1. Write the augmented matrix of the system.

2. Perform row operations to obtain a simplified form of the matrix.

3. Use back substitution to find the values of the variables. Example: Solve the following system of equations using the Gaussian elimination method:

• x + 2y = 4
• 2x + 3y = 7

Gaussian Elimination Method (continued)

1. Write the augmented matrix of the system:
   • [1 2 | 4] [2 3 | 7]

2. Perform row operations to obtain a simplified form of the matrix:
   • R2 = R2 - 2R1 [1 2 | 4] [0 -1 | -1]

3. Use back substitution to find the values of the variables:

   • -y = -1 y = 1

   • Substitute the value of y into equation (1) to find x: x + 2(1) = 4 x + 2 = 4 x = 2

Solution Verification

To verify the solutions obtained from the Gaussian elimination method, substitute the values of the variables into the original equations.

1. Substitute x = 2 and y = 1 into the first equation:
   • 2 + 2(1) = 4 2 + 2 = 4 4 = 4 (True)

2. Substitute x = 2 and y = 1 into the second equation:
   • 2(2) + 3(1) = 7 4 + 3 = 7 7 = 7 (True) Since both equations are true, the solution x = 2 and y = 1 is verified.

Matrix Inversion Method

The matrix inversion method for solving systems of linear equations involves the following steps:

1. Write the augmented matrix of the system.

2. Find the inverse of the coefficient matrix.

3. Multiply the inverse matrix with the constant matrix to obtain the solution matrix. Example: Solve the following system of equations using the matrix inversion method:

• x + 2y = 4
• 2x + 3y = 7

Matrix Inversion Method (continued)

1. Write the augmented matrix of the system:
   • [1 2 | 4] [2 3 | 7]

2. Find the inverse of the coefficient matrix:

   • [1 2]^(-1) = 1/(-1) [3 -2] [2 3] [-2 1/(-1)]

   • [1 2]^(-1) = [-3 2] [2 3] [2 -1]

3. Multiply the inverse matrix with the constant matrix to obtain the solution matrix:
   • [-3 2] [4] = [2] [2 -1] [7] [-1]
   • The solution matrix is [x] = [2] [y] [-1]

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21. Matrix Addition and Subtraction

• Matrices can be added or subtracted if they have the same dimensions.
• To add or subtract matrices, simply add or subtract the corresponding elements.
• Example:
  • A = [[1, 2], [3, 4]] B = [[5, 6], [7, 8]] A + B = [[1+5, 2+6], [3+7, 4+8]] = [[6, 8], [10, 12]]

22. Scalar Multiplication

• A matrix can be multiplied by a scalar value by multiplying each element of the matrix by the scalar.
• The scalar multiplication operation does not change the dimensions of the matrix.
• Example:
  • A = [[1, 2], [3, 4]] k = 2 kA = [[21, 22], [23, 24]] = [[2, 4], [6, 8]]

23. Matrix Multiplication

• Matrix multiplication is only possible if the number of columns in the first matrix is equal to the number of rows in the second matrix.
• To multiply two matrices, we multiply each element of the row of the first matrix with the corresponding element of the column of the second matrix and sum the products.
• Example:
  • A = [[1, 2], [3, 4]] B = [[5, 6], [7, 8]] AB = [[(15 + 27), (16 + 28)], [(35 + 47), (36 + 48)]] = [[19, 22], [43, 50]]

24. Identity Matrix

• The identity matrix, denoted as I, is a special square matrix where all the elements on its main diagonal are equal to 1 and all other elements are equal to 0.
• The identity matrix has the property that when multiplied with any matrix A, the result is always A.
• Example:
  • I = [[1, 0, 0], [0, 1, 0], [0, 0, 1]] A = [[2, 3, 4], [5, 6, 7], [8, 9, 10]] IA = AI = A

25. Transpose of a Matrix

• The transpose of a matrix is obtained by interchanging its rows with columns.
• The transpose of a matrix A is denoted as A^T.
• Example:
  • A = [[1, 2, 3], [4, 5, 6]] A^T = [[1, 4], [2, 5], [3, 6]]

26. Determinant of a Matrix

• The determinant of a square matrix A, denoted as |A| or det(A), is a scalar value that can be calculated using various methods such as cofactor expansion or row reduction.
• The determinant provides important information about the matrix, such as its invertibility and the nature of its solutions when used in solving systems of equations.
• Example:
  • A = [[1, 2], [3, 4]] |A| = 14 - 23 = 4 - 6 = -2

27. Properties of Determinants

• The determinant of a matrix A is not affected by scalar multiplication of A.
• If two rows or columns of a matrix A are interchanged, the determinant changes sign.
• The determinant of a matrix A is equal to the determinant of its transpose.
• If one row or column of a matrix A is multiplied by k, the determinant is multiplied by k.
• Example:
  • A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]] |A| = -0 (by cofactor expansion) Transpose of A: [[1, 4, 7], [2, 5, 8], [3, 6, 9]] |Transpose of A| = 0 (same as |A|)

28. Inverse of a Matrix

• The inverse of a square matrix A, denoted as A^(-1), is a matrix such that when multiplied with A, the result is the identity matrix I.
• Not all matrices have inverses; a matrix must be non-singular or have a non-zero determinant to have an inverse.
• Example:
  • A = [[2, 3], [4, 5]] A^(-1) = [[-5/2, 3/2], [2, -1]]

29. Properties of Matrix Inverses

• The inverse of a matrix A is unique if it exists.
• If A^(-1) exists and A is invertible, then (A^(-1))^(-1) = A.
• If A and B are invertible matrices, then AB is also invertible and (AB)^(-1) = B^(-1)A^(-1).
• If A^(-1) exists, then (A^T)^(-1) also exists and (A^T)^(-1) = (A^(-1))^T.
• Example:
  • A = [[1, 2], [3, 4]] B = [[5, 6], [7, 8]] A^(-1) = [[-2, 1], [3/2, -1/2]] B^(-1) = [[-4, 3], [3.5, -2.5]] AB = [[5, 12], [21, 32]] AB^(-1) = [[-32, 21], [21.5, -13.5]]

30. Solving Systems of Equations using Matrices

• Matrices can be used to solve systems of linear equations by representing the system in augmented matrix form.
• Gaussian elimination or matrix inversion methods can be applied to obtain the solution.
• Example:
  • Solve the system of equations:
    • 2x + 3y = 8
    • 4x + 5y = 14