Matrix and Determinant

• Problem on inverse and adjoint of a matrix

Inverse of a Matrix

• The inverse of a matrix A is denoted as A^-1
• If A is a square matrix of order n and AB = BA = I_n, then B is the inverse of A
• A matrix has an inverse if and only if its determinant is non-zero

Determinant of a Matrix

• The determinant of a square matrix A is denoted as |A|
• It is a scalar value calculated from the elements of the matrix
• Determinants are only defined for square matrices

Properties of Inverse

• If A is invertible, then A^-1 is unique
• (AB)^-1 = B^-1A^-1
• (A^-1)^-1 = A
• (kA)^-1 = (1/k)A^-1 (where k is a non-zero scalar)
• (A^T)^-1 = (A^-1)^T

Problem 1

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

Solution 1

First, calculate the determinant of A: |A| = (2 * 3) - (4 * 1) = 6 - 4 = 2 Since the determinant is non-zero, A has an inverse. Next, find the adjoint of A: A_adj = [3, -1; -4, 2] Finally, divide the adjoint of A by the determinant: A^-1 = (1/2) * [3, -1; -4, 2]

Problem 2

Find the inverse of the matrix B = [5, 2, 1; 4, 3, 1; 2, 1, 1]

Solution 2

First, calculate the determinant of B: |B| = 5 * (3 * 1 - 1 * 1) - 2 * (4 * 1 - 1 * 1) + 1 * (4 * 1 - 3 * 1) |B| = 5 * (3 - 1) - 2 * (4 - 1) + 1 * (4 - 3) |B| = 5 * 2 - 2 * 3 + 1 * 1 |B| = 10 - 6 + 1 |B| = 5 Since the determinant is non-zero, B has an inverse. Next, find the adjoint of B: B_adj = [2, -1, 1; -2, 4, -1; -1, 1, -1] Finally, divide the adjoint of B by the determinant: B^-1 = (1/5) * [2, -1, 1; -2, 4, -1; -1, 1, -1]

Problem 3

Find the inverse of the matrix C = [1, 2; 3, 4]

Solution 3

First, calculate the determinant of C: |C| = (1 * 4) - (3 * 2) = 4 - 6 = -2 Since the determinant is non-zero, C has an inverse. Next, find the adjoint of C: C_adj = [4, -2; -3, 1] Finally, divide the adjoint of C by the determinant: C^-1 = (1/-2) * [4, -2; -3, 1]

Matrix and Determinant

• Problem on inverse and adjoint of a matrix

Example 1

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

• Determine the determinant of A
• Find the adjoint matrix of A
• Divide the adjoint matrix by the determinant to find the inverse

Example 2

Find the inverse of the matrix B = [9, 2; 7, 8]

• Calculate the determinant of B
• Find the adjoint matrix of B
• Divide the adjoint matrix by the determinant to find the inverse

Example 3

Find the inverse of the matrix C = [6, 2; 3, 4]

• Determine the determinant of C
• Find the adjoint matrix of C
• Divide the adjoint matrix by the determinant to find the inverse

Properties of Determinants

• If A and B are two square matrices of the same order, then |AB| = |A||B|
• If A is a square matrix and c is a scalar, then |cA| = cⁿ|A|, where n is the order of the matrix
• If A and B are two square matrices of the same order, then |A+B| &

##8805; |A| + |B|

Cramer’s Rule

• Cramer’s rule is a formula used to solve a system of linear equations using determinants
• For a system of linear equations Ax = b, where A is the coefficient matrix, x is the vector of variables, and b is the vector of constants, we can use Cramer’s rule to find the values of x₁, x₂, …, x_n
• x₁ = |B₁| / |A|, where B₁ is obtained by replacing the first column of A with b
• x₂ = |B₂| / |A|, where B₂ is obtained by replacing the second column of A with b
• x_n = |B_n| / |A|, where B_n is obtained by replacing the nth column of A with b

Example 4

Solve the system of linear equations:

2x + 3y = 7

4x - 2y = 2

• Write the coefficient matrix A and the vector of constants b
• Calculate the determinant of A
• Calculate the determinants of matrices B₁ and B₂
• Solve for x and y using Cramer’s rule

Properties of Inverse

• If A is invertible, then A^-1 is unique
• (AB)^-1 = B^-1A^-1
• (A^-1)^-1 = A
• (kA)^-1 = (1/k)A^-1 (where k is a non-zero scalar)
• (A^T)^-1 = (A^-1)^T

Summary

• Inverse of a matrix is only defined for square matrices with non-zero determinants
• The determinant of a matrix is a scalar value calculated from its elements
• Inverse and adjoint of a matrix can be used to solve linear equations
• Cramer’s rule provides a method to solve systems of linear equations using determinants

Questions?

• Do you have any questions on the topic of inverse and adjoint of a matrix?
• Let’s solve some practice problems to reinforce the concepts we’ve learned

Matrix and Determinant - Problem on inverse and adjoint of a matrix

• Problem 4: Find the inverse of the matrix D = [3, 1, 2; 5, 2, 4; 1, 0, 2]
• Solution 4:
  • Calculate the determinant of D
  • Find the adjoint matrix of D
  • Divide the adjoint matrix by the determinant to find the inverse

Matrix and Determinant - Problem on inverse and adjoint of a matrix

• Problem 5: Find the inverse of the matrix E = [4, 6, 1; 3, 2, 5; 7, 1, 3]
• Solution 5:
  • Calculate the determinant of E
  • Find the adjoint matrix of E
  • Divide the adjoint matrix by the determinant to find the inverse

Matrix and Determinant - Property of Determinants

• Property 1: If two rows or columns of a matrix are interchanged, the determinant changes its sign.
• Property 2: If any row (or column) of a matrix is multiplied by a non-zero scalar, the determinant is multiplied by the same scalar.
• Property 3: If a row (or column) of a matrix is replaced by the sum of itself and the corresponding row (or column) of another matrix, the determinant remains unchanged.

Matrix and Determinant - Property of Determinants

• Property 4: The determinant of the identity matrix I_n, where n is the order of the square matrix, is 1.
• Property 5: If two rows or columns of a matrix are proportional, the determinant of the matrix is 0.
• Property 6: If any two rows or columns of a matrix are proportional, the value of the determinant is zero.

Matrix and Determinant - Cramer’s Rule

• Cramer’s rule can be used to solve a system of linear equations with n variables.
• For a system of equations Ax = b, where A is the coefficient matrix, x is the vector of variables, and b is the vector of constants, Cramer’s rule states that:
  • x₁ = |A₁| / |A|
  • x₂ = |A₂| / |A|
  • …
  • x_n = |A_n| / |A|

Matrix and Determinant - Cramer’s Rule Example

Solve the following system of equations using Cramer’s rule:

3x + 2y + z = 12

2x - y + 3z = 6 x + 2y - z = 4

• Write the coefficient matrix A and the vector of constants b
• Calculate the determinant of A
• Calculate the determinants of matrices A₁, A₂, and A₃
• Solve for x, y, and z using Cramer’s rule

Matrix and Determinant - Cramer’s Rule Example

Solve the following system of equations using Cramer’s rule:

2x - 3y + z = 5

3x + 2y - z = 7 x + 3y + 2z = 8

• Write the coefficient matrix A and the vector of constants b
• Calculate the determinant of A
• Calculate the determinants of matrices A₁, A₂, and A₃
• Solve for x, y, and z using Cramer’s rule

Matrix and Determinant - Summary

• Inverse of a matrix is only defined for square matrices with non-zero determinants
• The determinant of a matrix is a scalar value calculated from its elements
• Properties of determinants can be used to simplify calculations and transformations on matrices
• Cramer’s rule provides a method to solve systems of linear equations using determinants

Matrix and Determinant - Questions?

• Do you have any questions on the topic of inverse and adjoint of a matrix, determinants, or Cramer’s rule?
• Let’s solve some practice problems to reinforce the concepts we’ve learned

Matrix and Determinant - Thank You!

• Thank you for attending this lecture on matrix and determinant
• If you have any further questions, feel free to reach out to me
• Stay tuned for more topics in our upcoming lectures!