Determinants - Evaluation Of Inverse Of A Square Matrix Using Determinants And Elementary Transformations

Determinants - Evaluation of inverse of a square matrix using determinants and elementary transformations

Inverse of a square matrix:
- Definition: The inverse of a square matrix A is denoted by A^-1.
- The inverse of A exists only if the determinant of A is not zero.
- A * A^-1 = A^-1 * A = I, where I is the identity matrix.
Determinants:
- Definition: A determinant is a scalar value that can be calculated for a square matrix.
- Determinant is denoted by |A| or det(A).
- It determines the invertibility of a matrix.
- Determinants can be evaluated using elementary transformations.
Elementary transformations:
- Row Interchange: Interchanging the positions of two rows in a matrix.
- Row Scaling: Multiplying all the elements of a row by a non-zero scalar.
- Row Replacement: Adding or subtracting multiples of one row to another row.
Steps to evaluate the inverse of a matrix:
1. Find the determinant of the matrix A: |A|.
2. If |A| = 0, the matrix A does not have an inverse.
3. If |A| ≠ 0, proceed to find the adjoint of A.
4. The adjoint of A is obtained by taking the transpose of the cofactor matrix of A.
5. Divide the adjoint of A by the determinant of A to obtain the inverse of A: A^-1 = adj(A) / |A|.
Example:
- Consider the matrix A = [[3, -2], [1, 4]]:
- Step 1: Find the determinant |A| = (3 * 4) - (-2 * 1) = 14.
- Since |A| ≠ 0, the matrix A has an inverse.

Slide 11:

Steps to evaluate the inverse of a matrix (contd.): 6. Find the cofactor matrix of A, denoted by C.
- The cofactor of each element a_ij in A is given by C_ij = (-1)^(i+j) * M_ij, where M_ij is the determinant of the minor matrix obtained by removing the i-th row and j-th column from A.
1. Take the transpose of C to obtain the adjoint matrix, denoted by adj(A).
2. Divide adj(A) by |A| to obtain A^-1.
3. Simplify the resulting matrix, if necessary.
Example (contd.):
- Step 2: Finding the adjoint of A:
  - C = [[4, 1], [-2, 3]]
  - adj(A) = [[4, -2], [1, 3]]
- Step 3: Divide adj(A) by |A|:
  - A^-1 = adj(A) / |A| = [[4/14, -2/14], [1/14, 3/14]]
- Therefore, the inverse of matrix A = [[3, -2], [1, 4]] is A^-1 = [[4/14, -2/14], [1/14, 3/14]].

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Properties of inverse of a matrix:
1. If A is a square matrix and A^-1 exists, then (A^-1)^-1 = A.
2. If A and B are invertible square matrices of the same order, then AB is also invertible, and (AB)^-1 = B^-1 * A^-1.
3. If A is invertible, then (A^T)^-1 = (A^-1)^T.
4. If A is invertible, then the inverse of the transpose of A is the transpose of its inverse.
5. If A is invertible, then the inverse of the product of n copies of A is the product of n copies of A^-1.
Example:
- Suppose A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]].
- A^-1 = [[-2, 1], [3/2, -1/2]]
- B^-1 = [[-4, 3], [3.5, -2.5]]
- Therefore, (AB)^-1 = B^-1 * A^-1 = [[-4, 3], [3.5, -2.5]] * [[-2, 1], [3/2, -1/2]].
- Simplifying the multiplication gives us the inverse of AB.

Slide 13:

Properties of determinants:
1. For a square matrix A, if A^-1 exists, then |A| ≠ 0.
2. If A and B are square matrices of the same order, then |AB| = |A| * |B|.
3. For any square matrix A, |A^T| = |A|.
4. If A is an invertible square matrix, then |A^-1| = 1 / |A|.
5. If A is a triangular matrix, then |A| is equal to the product of the elements on the main diagonal.
Example:
- Consider the matrix A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]].
- |A| = 1 * (5 * 9 - 6 * 8) - 2 * (4 * 9 - 6 * 7) + 3 * (4 * 8 - 5 * 7).
- Simplifying the expression gives us the determinant of A.

Slide 14:

Application of determinants in solving systems of linear equations:
- Determinants can be used to determine whether a system of linear equations has a unique solution, no solution, or infinitely many solutions.
- If the determinant of the coefficient matrix is not zero, the system has a unique solution.
- If the determinant of the coefficient matrix is zero, the system may have no solution or infinitely many solutions.
- The determinant can also be used to find the values of unknowns in a system of linear equations.
Example:
- Consider the system of linear equations:
  - 2x + 3y = 7
  - 4x - 5y = 11
- The coefficient matrix is A = [[2, 3], [4, -5]], and the determinant |A| ≠ 0.
- Therefore, the system has a unique solution.

Slide 15:

Determinants of higher order matrices:
- The determinants of higher order matrices can be evaluated using expansion by minors.
- Expansion by minors involves selecting a row or column of the matrix and multiplying each element of that row or column with the determinant of the minor matrix obtained by deleting the row and column of the selected element.
- This process is repeated for each element in the selected row or column and the results are added or subtracted, depending on the position of the element.
Example:
- Consider the matrix A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]].
- We can evaluate the determinant by expanding along the first row:
  - |A| = 1 * |[[5, 6], [8, 9]]| - 2 * |[[4, 6], [7, 9]]| + 3 * |[[4, 5], [7, 8]]|.
- Simplifying each sub-determinant gives us the final value of |A|.

Slide 16:

Cramer’s Rule:
- Cramer’s Rule provides a method to solve a system of linear equations using determinants.
- It states that the ratio of the determinant of a matrix obtained by replacing the column of coefficients of one variable with the column of constants, to the determinant of the coefficient matrix, gives the value of that variable.
- This rule allows us to solve for each variable in the system of equations individually.
Example:
- Consider the system of linear equations:
  - 2x + 3y = 7
  - 4x - 5y = 11
- Using Cramer’s Rule, we can find the values of x and y by evaluating the determinants:
  - x = |[[7, 3], [11, -5]]| / |[[2, 3], [4, -5]]|
  - y = |[[2, 7], [4, 11]]| / |[[2, 3], [4, -5]]|
- Simplifying each determinant will give us the values of x and y.

Slide 17:

Limitations of determinant approach:
1. The determinant method is computationally expensive for large matrices.
2. The determinant method can be difficult to apply for matrices with symbolic entries.
3. The method gives insight into invertibility but may not provide a complete understanding of matrix properties.
4. The method relies on finding determinants, which may involve complex calculations.
Despite these limitations, the determinant approach is widely used in various areas of mathematics, such as linear algebra, differential equations, and physics.
It is important to understand the concept of determinants and their applications in order to solve problems related to matrices and systems of linear equations.

Slide 18:

Summary:
- Determinants are scalar values that can be calculated for square matrices.
- Determinants determine the invertibility of a matrix.
- The inverse of a matrix exists only if the determinant is not zero.
- Inverse of a matrix can be evaluated using determinants and elementary transformations.
- The inverse of a matrix satisfies specific properties.
- Determinants can be used to solve systems of linear equations and evaluate the values of unknowns.
- Cramer’s Rule provides a method to solve systems of equations using determinants.
Understanding these concepts and their applications can help in solving various mathematical problems and gaining a deeper understanding of linear algebra.

Slide 21:

Solving a system of three equations with three variables using determinants:
- Consider the system of equations:
  - 2x + y - z = 3
  - 3x - 2y + 4z = -2
  - x + 3y - 2z = 7
- We can represent this system using matrix form: AX = B, where A is the coefficient matrix, X is the unknown variables, and B is the constant matrix.
Steps to solve the system using determinants:
1. Calculate the determinant of matrix A, denoted by |A|.
2. Calculate the determinants of matrix B1, B2, and B3, where B1, B2, and B3 are obtained by replacing the first, second, and third columns of A with the column matrix B.
3. Find the values of x, y, and z using the formulas: x = |B1| / |A|, y = |B2| / |A|, and z = |B3| / |A|.
Example:
- Consider the system of equations:
  - 2x + y - z = 3
  - 3x - 2y + 4z = -2
  - x + 3y - 2z = 7
- We can represent this system using matrix form:
  - A = [[2, 1, -1], [3, -2, 4], [1, 3, -2]]
  - X = [[x], [y], [z]]
  - B = [[3], [-2], [7]]

Slide 22:

Solving a system of three equations with three variables using determinants (contd.):
- Step 1: Calculate the determinant |A| = 2(4(-2) - 3(-2)) - 1(3(-2) - 1(4)) + (-1)(3(-2) - 1(3))
- Step 2: Calculate the determinants |B1|, |B2|, and |B3| using the formulas:
  - |B1| = 3(4(-2) - 3(7)) - (-1)(-2(7) - 3(3))
  - |B2| = 2(-2(7) - 3(7)) - (-1)(3(-2) - 1(7))
  - |B3| = 2(3(7) - 4(-2)) - 1(3(7) - 1(3))
- Step 3: Find the values of x, y, and z using the formulas:
  - x = |B1| / |A|
  - y = |B2| / |A|
  - z = |B3| / |A|
Example:
- Consider the system of equations:
  - 2x + y - z = 3
  - 3x - 2y + 4z = -2
  - x + 3y - 2z = 7
- We have already calculated |A|, |B1|, |B2|, and |B3| in the previous slide.
- Now, we can find the values of x, y, and z using the formulas:
  - x = |B1| / |A|
  - y = |B2| / |A|
  - z = |B3| / |A|

Slide 23:

Example (contd.):
- Substitute the calculated determinants into the formulas to find the values of x, y, and z:
  - x = |B1| / |A|
  - y = |B2| / |A|
  - z = |B3| / |A|
- Simplify the expressions to obtain the final values of x, y, and z.
Therefore, by evaluating the determinants, we can find the values of the unknown variables in a system of linear equations.

Slide 24:

Cofactor expansion method:
- In addition to using elementary transformations, determinants can be evaluated using the cofactor expansion method.
- This method involves expanding the determinant along a row or column by using the cofactors of the elements in that row or column.
Cofactor expansion along the first row:
- Consider a square matrix A.
- The determinant of A can be expanded along the first row as:
  - |A| = a_11 * C_11 - a_12 * C_12 + … + (-1)^(n+1) * a_1n * C_1n, where C_ij is the cofactor of element a_ij.
Example:
- Consider the matrix A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]].
- We can expand the determinant of A along the first row using the cofactors:
  - |A| = 1 * C_11 - 2 * C_12 + 3 * C_13.

Slide 25:

Cofactor expansion method (contd.):
- The cofactor of each element a_ij in A is given by C_ij = (-1)^(i+j) * |M_ij|, where |M_ij| is the determinant of the minor matrix obtained by deleting the i-th row and j-th column from A.
Example (contd.):
- Let’s find the cofactors of each element in A:
  - C_11 = (-1)^(1+1) * |M_11|
  - C_12 = (-1)^(1+2) * |M_12|
  - C_13 = (-1)^(1+3) * |M_13|
  - C_21 = (-1)^(2+1) * |M_21|
  - C_22 = (-1)^(2+2) * |M_22|
  - C_23 = (-1)^(2+3) * |M_23|
  - C_31 = (-1)^(3+1) * |M_31|
  - C_32 = (-1)^(3+2) * |M_32|
  - C_33 = (-1)^(3+3) * |M_33|
Each |M_ij| can be evaluated by finding the determinants of the minor matrices obtained by removing the corresponding row and column.

Slide 26:

Cofactor expansion method (contd.):
- Let’s evaluate each |M_ij|:
  - |M_11| = |[[5, 6], [8, 9]]|
  - |M_12| = |[[4, 6], [7, 9]]|
  - |M_13| = |[[4, 5], [7, 8]]|
  - |M_21| = |[[2, 3], [8, 9]]|
  - |M_22| = |[[1, 3], [7, 9]]|
  - |M_23| = |[[1, 2], [7, 8]]|
  - |M_31| = |[[2, 3], [5, 6]]|
  - |M_32| = |[[1, 3], [4, 6]]|
  - |M_33| = |[[1, 2], [4, 5]]|
We can now substitute the values of the cofactors and solve for |A| using the cofactor expansion method.

Slide 27:

Cofactor expansion method (contd.):
- Substituting the values of the cofactors, we get:
  - |A| = 1 * C_11 - 2 * C_12 + 3