Determinants - Polynomial Trick For Calculating Inverse

Determinants - Polynomial trick for calculating inverse

We know that the inverse of a matrix A is denoted by A^(-1)
To calculate the inverse of a matrix using the polynomial trick, we follow these steps:
1. Find the determinant of the given matrix A.
2. If the determinant is non-zero, proceed to the next steps. Otherwise, the matrix is not invertible.
3. Find the adjugate of the given matrix A.
4. Multiply the adjugate by the reciprocal of the determinant to get the inverse matrix.

Example: Find the inverse of the matrix A = [2 1; 4 3]

Step 1: Determinant of A = (2 * 3) - (1 * 4) = 6 - 4 = 2
Step 2: Since the determinant is non-zero, we proceed.
Step 3: Adjugate of A = [3 -1; -4 2]
Step 4: Inverse of A = (1/2) * [3 -1; -4 2] = [3/2 -1/2; -2 1]

Properties of determinants

Determinant of a square matrix A is denoted by |A|.
Some important properties of determinants are:
1. |A^T| = |A|
2. |AB| = |A| * |B|
3. |A^(-1)| = 1/|A|
4. If A is invertible, |A^(-1)| = 1/|A|

Example: Find the determinant of the matrix A = [1 2 3; 4 5 6; 7 8 9]

We can calculate the determinant of a 3x3 matrix using the formula: |A| = a(ei - fh) - b(di - fg) + c(dh - eg)
Substituting the values from matrix A, we have: |A| = 1(59 - 68) - 2(49 - 67) + 3(48 - 57) = 1(45 - 48) - 2(36 - 42) + 3(32 - 35) = 1*(-3) - 2*(-6) + 3*(-3) = -3 + 12 - 9 = 0

Cramer’s Rule

Cramer’s Rule is a method for solving a system of linear equations using determinants.
For a system of equations in the form AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix, the solution can be found using the following formulas: X1 = |B1 A2 A3| / |A| X2 = |A1 B2 A3| / |A| X3 = |A1 A2 B3| / |A|
Here, |A| represents the determinant of the coefficient matrix A, and each Xi represents the value of the ith variable.

Example: Solve the system of equations using Cramer’s Rule:

2x + 3y - 4z = 10
5x - 6y + z = -2
x + 2y - 3z = 4
Coefficient matrix A = [2 3 -4; 5 -6 1; 1 2 -3]
Constant matrix B = [10; -2; 4]
Using Cramer’s Rule, we can find the values of x, y, and z as follows: X1 = |10 3 -4| / |-20 + 18 + 12| = 10 / 10 = 1

The Laplace Expansion Theorem

The Laplace Expansion Theorem states that the determinant of a square matrix A can be calculated by expanding along any row or column.
To calculate the determinant using the Laplace Expansion Theorem, follow these steps:
1. Choose a row or column to expand along.
2. For each element in that row or column, calculate the determinant of the minor matrix.
3. Multiply each determinant by the corresponding element.
4. Add or subtract the products alternately to get the final determinant.

Example: Find the determinant of the matrix A = [1 2 -1; -3 4 2; 2 1 3]

Let’s expand along the first row.
Minor matrix corresponding to the element 1: [4 2; 1 3]
Determinant of minor matrix 1 = (4 * 3) - (2 * 1) = 12 - 2 = 10
Minor matrix corresponding to the element 2: [-3 2; 2 3]
Determinant of minor matrix 2 = (-3 * 3) - (2 * 2) = -9 - 4 = -13
Minor matrix corresponding to the element -1: [-3 4; 2 1]
Determinant of minor matrix -1 = (-3 * 1) - (4 * 2) = -3 - 8 = -11
Final determinant = (1 * 10) - (2 * -13) + (-1 * -11) = 10 + 26 + 11 = 47

Solving Systems of Equations using Determinants

We can solve a system of linear equations using determinants.
For a system of equations AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix, the solution can be found using the following formula: X = A^(-1) * B
Here, A^(-1) represents the inverse of matrix A.

Example: Solve the system of equations using determinants:

2x + 3y = 13
4x - 7y = -5
Coefficient matrix A = [2 3; 4 -7]
Constant matrix B = [13; -5]
To find X, we need to calculate the inverse of matrix A.
We can use the polynomial trick to find A^(-1).

Slide 11: Determinants - Polynomial trick for calculating inverse

We know that the inverse of a matrix A is denoted by A^(-1)
To calculate the inverse of a matrix using the polynomial trick, we follow these steps:
- Find the determinant of the given matrix A.
- If the determinant is non-zero, proceed to the next steps. Otherwise, the matrix is not invertible.
- Find the adjugate of the given matrix A.
- Multiply the adjugate by the reciprocal of the determinant to get the inverse matrix.

Slide 12: Example - Find inverse of matrix A

Consider the matrix A = [2 1; 4 3]
Step 1: Determinant of A = (2 * 3) - (1 * 4) = 6 - 4 = 2
Step 2: Since the determinant is non-zero, we proceed.
Step 3: Adjugate of A = [3 -1; -4 2]
Step 4: Inverse of A = (1/2) * [3 -1; -4 2] = [3/2 -1/2; -2 1]

Slide 13: Properties of determinants

Determinant of a square matrix A is denoted by |A|.
Some important properties of determinants are:
- |A^T| = |A|
- |AB| = |A| * |B|
- |A^(-1)| = 1/|A|
- If A is invertible, |A^(-1)| = 1/|A|

Slide 14: Example - Calculate determinant of matrix A

Consider the matrix A = [1 2 3; 4 5 6; 7 8 9]
Using the formula for a 3x3 matrix determinant:
- |A| = a(ei - fh) - b(di - fg) + c(dh - eg)
Substituting the values from matrix A, we have:
- |A| = 1(59 - 68) - 2(49 - 67) + 3(48 - 57) = 1(45 - 48) - 2(36 - 42) + 3(32 - 35) = 1*(-3) - 2*(-6) + 3*(-3) = -3 + 12 - 9 = 0

Slide 15: Cramer’s Rule

Cramer’s Rule is a method for solving a system of linear equations using determinants.
For a system of equations in the form AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix, the solution can be found using the following formulas:
- X1 = |B1 A2 A3| / |A|
- X2 = |A1 B2 A3| / |A|
- X3 = |A1 A2 B3| / |A|
Here, |A| represents the determinant of the coefficient matrix A, and each Xi represents the value of the ith variable.

Slide 16: Example - Solve system of equations using Cramer’s Rule

Consider the system of equations:
- 2x + 3y - 4z = 10
- 5x - 6y + z = -2
- x + 2y - 3z = 4
Coefficient matrix A = [2 3 -4; 5 -6 1; 1 2 -3]
Constant matrix B = [10; -2; 4]
Using Cramer’s Rule, we can find the values of x, y, and z as follows:
- X1 = |10 3 -4| / |-20 + 18 + 12| = 10 / 10 = 1

Slide 17: The Laplace Expansion Theorem

The Laplace Expansion Theorem states that the determinant of a square matrix A can be calculated by expanding along any row or column.
To calculate the determinant using the Laplace Expansion Theorem, follow these steps:
- Choose a row or column to expanding along.
- For each element in that row or column, calculate the determinant of the minor matrix.
- Multiply each determinant by the corresponding element.
- Add or subtract the products alternately to get the final determinant.

Slide 18: Example - Find determinant using Laplace Expansion Theorem

Consider the matrix A = [1 2 -1; -3 4 2; 2 1 3]
Let’s expand along the first row.
Minor matrix corresponding to element 1: [4 2; 1 3]
Determinant of minor matrix 1 = (4 * 3) - (2 * 1) = 12 - 2 = 10
Minor matrix corresponding to element 2: [-3 2; 2 3]
Determinant of minor matrix 2 = (-3 * 3) - (2 * 2) = -9 - 4 = -13
Minor matrix corresponding to element -1: [-3 4; 2 1]
Determinant of minor matrix -1 = (-3 * 1) - (4 * 2) = -3 - 8 = -11
Final determinant = (1 * 10) - (2 * -13) + (-1 * -11) = 10 + 26 + 11 = 47

Slide 19: Solving Systems of Equations using Determinants

We can solve a system of linear equations using determinants.
For a system of equations AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix, the solution can be found using the following formula:
- X = A^(-1) * B
Here, A^(-1) represents the inverse of matrix A.

Slide 20: Example - Solving a system of equations using determinants

Consider the system of equations:
- 2x + 3y = 13
- 4x - 7y = -5
Coefficient matrix A = [2 3; 4 -7]
Constant matrix B = [13; -5]
To find X, we need to calculate the inverse of matrix A using the polynomial trick.
Multiply the inverse of matrix A with matrix B:
- X = A^(-1) * B

Slide 21: Calculating Determinants

Determinants can be calculated using various methods, such as Laplace Expansion or Row Operations.
The choice of method depends on the size and nature of the matrix.
For matrices bigger than 3x3, row operations or software tools are commonly used.
Laplace Expansion is suitable for smaller matrices.
Different methods can yield the same determinant value.

Slide 22: Laplace Expansion Method

Laplace Expansion is a technique to calculate determinants by expanding along a row or column.
The expansion creates a series of smaller matrices called minors.
Each minor’s determinant is evaluated by following the same process recursively until a base case is reached.
The determinants of minors are combined using alternating signs and multiplied by the corresponding element of the original matrix.

Slide 23: Example - Laplace Expansion

Consider the matrix A = [3 1 2; 5 2 4; 1 0 3]
Expand along the first row:
- Minor 1: [2 4; 0 3] - Determinant = (2 * 3) - (0 * 4) = 6
- Minor 2: [5 4; 1 3] - Determinant = (5 * 3) - (1 * 4) = 15 - 4 = 11
- Minor 3: [5 2; 1 0] - Determinant = (5 * 0) - (1 * 2) = 0 - 2 = -2
Calculate determinant: (3 * 6) - (1 * 11) + (2 * -2) = -12 - 11 - 4 = -27

Slide 24: Properties of Determinants

Determinants have several important properties:
1. The determinant of the identity matrix is 1.
2. Swapping rows or columns negates the determinant.
3. Multiplying a row or column by a scalar multiplies the determinant by the same scalar.
4. Adding or subtracting a multiple of one row or column to another row or column does not change the determinant.
5. If two rows or two columns of a matrix are identical, the determinant is 0.

Slide 25: Unique Solutions and Inverse Matrices

A square matrix has a unique solution if and only if its determinant is non-zero.
If a matrix has a non-zero determinant, it is said to be invertible or non-singular.
The inverse of an invertible square matrix A is denoted as A^(-1).
The inverse matrix satisfies the property A * A^(-1) = A^(-1) * A = I, where I is the identity matrix.

Slide 26: Calculating Inverse of a Matrix

The inverse of a matrix can be calculated using various methods, such as the polynomial trick or row operations.
The polynomial trick is a systematic approach that results in an explicit formula for the inverse.
It involves finding the determinant of the matrix, calculating the adjugate matrix, and then multiplying by the reciprocal of the determinant.

Slide 27: Example - Calculating Inverse

Consider the matrix A = [2 1; 4 3]
Step 1: Determine the determinant of A = (2 * 3) - (1 * 4) = 6 - 4 = 2
Step 2: Calculate the adjugate of A = [3 -1; -4 2]
Step 3: Multiply the adjugate by the reciprocal of the determinant:
- Inverse of A = (1/2) * [3 -1; -4 2] = [3/2 -1/2; -2 1]

Slide 28: Properties of Inverse Matrices

Inverse matrices have several important properties:
1. The inverse of the inverse of a matrix is the original matrix: (A^(-1))^(-1) = A.
2. The inverse of the product of two matrices is the product of their inverses in reverse order: (AB)^(-1) = B^(-1) * A^(-1).
3. The inverse of a scalar multiple of a matrix is the reciprocal of the scalar times the inverse of the matrix: (kA)^(-1) = (1/k) * A^(-1).
4. If a matrix is not invertible (determinant = 0), it does not have an inverse.

Slide 29: Solving Systems Using Inverse Matrices

Given a system of linear equations AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix, we can find the solution using the inverse matrix.
Multiply both sides of the equation by the inverse of A: X = A^(-1) * B.
The solution X will contain the values of the variables in the system.

Slide 30: Example - Solving a System Using Inverse Matrices

Consider the system of equations:
- 2x + y = 5
- 3x - 2y = -1
Coefficient matrix A = [