Determinants - Inverse of a matrix

• Definition of a matrix
• Definition of a determinant
• Properties of determinants
  • Scalar multiplication property
  • Row/column operations property
  • Transposition property
• Inverse of a matrix
  • Definition of inverse
  • Conditions for existence of inverse
  • Finding inverse using adjugate method
  • Finding inverse using elementary row operations
• Examples:
  • Finding the determinant of a matrix
  • Finding the inverse of a matrix
• Application of inverse matrices in solving systems of equations
• Recap and summary

Definition of a Matrix

• A matrix is a rectangular array of numbers, symbols or expressions arranged in rows and columns.
• It is denoted by a capital letter and its entries are denoted by lowercase letters or subscripts.

Definition of a Determinant

• The determinant of a square matrix is a numerical value that is uniquely associated with that matrix.
• It is denoted by det(A) or |A|.
• The determinant of a 2x2 matrix A = [[a, b], [c, d]] is calculated as ad - bc.

Properties of Determinants

• Scalar multiplication property:
  • If a matrix A has determinant |A|, then the determinant of kA is k^n * |A|, where n is the order of matrix A.
• Row/column operations property:
  • If two rows or columns of a matrix A are interchanged, then the determinant of A changes its sign.
  • If k times a row or column of A is multiplied by a scalar, then the determinant of A is multiplied by k.
• Transposition property:
  • If B is the transpose of A, then the determinant of B is equal to the determinant of A.

Inverse of a Matrix

• The inverse of a matrix A, denoted as A^(-1), is a matrix that, when multiplied with A, gives the identity matrix I.
• Only square matrices can have an inverse.

Conditions for Existence of Inverse

• A matrix A has an inverse A^(-1) if and only if its determinant |A| is non-zero.

Finding Inverse using Adjugate Method

• Find the determinant |A| of the matrix A.
• Find the adjugate of A denoted by adj(A).
• Determine the inverse A^(-1) using the formula A^(-1) = adj(A) / |A|.

Finding Inverse using Elementary Row Operations

• Form the augmented matrix [A | I], where I is the identity matrix.
• Perform row operations on [A | I] to transform A into the identity matrix.
• The transformed matrix [I | A^(-1)] will be the inverse of A.

Examples

Example 1: Find the determinant of the matrix A = [[2, 3], [4, 5]]. Solution: |A| = 25 - 34 = -2 Example 2: Find the inverse of the matrix B = [[2, 1], [3, 4]]. Solution: Step 1: |B| = 24 - 13 = 5 Step 2: adj(B) = [[4, -1], [-3, 2]] Step 3: B^(-1) = adj(B) / |B| = [[4/5, -1/5], [-3/5, 2/5]]

Application of Inverse Matrices

• Inverse matrices are used to solve systems of linear equations.
• If A is a matrix of coefficients and X is a matrix of variables, then the solution can be obtained by multiplying A^(-1) with X.
• The solution matrix X = A^(-1) * B, where B is the matrix of constants.

Recap and Summary

• A matrix is a rectangular array of numbers, denoted by a capital letter.
• The determinant of a matrix is a numerical value associated with it, denoted by |A|.
• Properties of determinants include scalar multiplication, row/column operations, and transposition.
• The inverse of a matrix A is denoted as A^(-1) and exists if |A| is non-zero.
• Inverse can be found using adjugate method or elementary row operations.
• Inverse matrices are useful in solving systems of equations.
• Understanding determinants and inverses is essential for further topics in linear algebra.

11. Determinants - Inverse of a matrix

• Example 3: Find the determinant of the matrix C = [[3, -2, 1], [4, 0, -3], [-1, 2, 5]]. Solution: |C| = 3 * (0 * 5 - 2 * (-3)) - (-2) * (4 * 5 - (-3) * (-1)) + 1 * (4 * 2 - 0 * (-1)) = 78
• Example 4: Find the inverse of the matrix D = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]. Solution: Step 1: |D| = 1 * (1 * 0 - 2 * 6) - 2 * (0 * 0 - 5 * 6) + 3 * (0 * 6 - 5 * 1) = -61 Step 2: adj(D) = [[-6, 18, -12], [30, -15, -5], [-12, 3, 2]] Step 3: D^(-1) = adj(D) / |D| = [[6/61, -18/61, 12/61], [-30/61, 15/61, 5/61], [12/61, -3/61, 2/61]]
• The inverse of a matrix is unique and can be verified by multiplying the matrix with its inverse to obtain the identity matrix.
• If a matrix A does not have an inverse (|A| = 0), it is called a singular matrix.
• The inverse of a diagonal matrix is obtained by taking the reciprocal of each diagonal element.

12. Inverse of a Matrix - Elementary Row Operations

• Elementary row operations include:
  1. Interchanging two rows.
  2. Multiplying a row by a non-zero scalar.
  3. Adding a multiple of one row to another row.
• Let’s consider an example to find the inverse of a matrix using elementary row operations.
• Example 5: Find the inverse of the matrix E = [[2, 3], [1, 4]] using elementary row operations. Solution: Step 1: Perform row operations on [E | I]. Multiply R2 by -2 and add it to R1 to get [1, 0 | -1, 2]. Multiply R1 by -1/5 to obtain the identity matrix on the left side. Step 2: The inverse of E is [0, -1/5 | 1/5, 2/5].
• Inverse matrices can be used to solve systems of linear equations efficiently.
• The matrix equation AX = B, where A is a matrix of coefficients, X is a matrix of variables, and B is a matrix of constants, can be solved using the inverse of A.
• Example 6: Solve the system of equations: 2x + 3y = 5 x + 4y = 10 Solution: The coefficient matrix A = [[2, 3], [1, 4]], variable matrix X = [[x], [y]], and constant matrix B = [[5], [10]]. Multiply both sides of the equation AX = B by A^(-1) to obtain X. Thus, X = [[-8], [7]] is the solution to the system of equations.

Slide 21

• The determinant of a matrix C = [[3, -2, 1], [4, 0, -3], [-1, 2, 5]] is 78
• Explaining the calculation step by step:
  • |C| = 3 * (0 * 5 - 2 * (-3)) - (-2) * (4 * 5 - (-3) * (-1)) + 1 * (4 * 2 - 0 * (-1))
  • |C| = 3 * (0 + 6) - (-2) * (20 + 3) + 1 * (8)
  • |C| = 3 * 6 - (-2) * 23 + 8
  • |C| = 18 + 46 + 8
  • |C| = 78

Slide 22

• Finding the inverse of the matrix D = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]
• Step 1: Calculate the determinant of D
  • |D| = 1 * (1 * 0 - 2 * 6) - 2 * (0 * 0 - 5 * 6) + 3 * (0 * 6 - 5 * 1)
  • |D| = 0 - 2 * (-30) + 3 * (-5)
  • |D| = 0 + 60 - 15
  • |D| = 45
• Step 2: Find the adjugate of D, denoted as adj(D)
  • adj(D) = [[-6, 18, -12], [30, -15, -5], [-12, 3, 2]]
• Step 3: Calculate the inverse of D using the formula D^(-1) = adj(D) / |D|
  • D^(-1) = [[-6/45, 18/45, -12/45], [30/45, -15/45, -5/45], [-12/45, 3/45, 2/45]]

Slide 23

• Inverse matrices are useful for solving systems of linear equations efficiently
• Consider the system of equations:
  • 2x + 3y = 5
  • x + 4y = 10
• The coefficient matrix A = [[2, 3], [1, 4]]
• The variable matrix X = [[x], [y]]
• The constant matrix B = [[5], [10]]
• Multiply both sides of the equation AX = B by A^(-1) to obtain X
• The inverse of A is [[0, -1/5], [1/5, 2/5]]
• X = A^(-1) * B
• X = [[0, -1/5], [1/5, 2/5]] * [[5], [10]]
• X = [[-8], [7]]
• Thus, the solution to the given system of equations is x = -8, y = 7

Slide 24

• A matrix that does not have an inverse is called a singular matrix
• Singular matrices have a determinant equal to zero
• If the determinant of a matrix A is zero, then A^(-1) does not exist
• The inverse of a matrix is unique
• The uniqueness can be verified by multiplying the matrix with its inverse to obtain the identity matrix

Slide 25

• The inverse of a diagonal matrix D is obtained by taking the reciprocal of each diagonal element
• Example:
  • Consider the diagonal matrix F = [[2, 0, 0], [0, -3, 0], [0, 0, 4]]
  • The inverse of F, F^(-1), is [[1/2, 0, 0], [0, -1/3, 0], [0, 0, 1/4]]
  • Verification: F * F^(-1) = [[2, 0, 0], [0, -3, 0], [0, 0, 4]] * [[1/2, 0, 0], [0, -1/3, 0], [0, 0, 1/4]] = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]

Slide 26

• The inverse of a matrix can also be found using elementary row operations
• Elementary row operations include:
  1. Interchanging two rows
  2. Multiplying a row by a non-zero scalar
  3. Adding a multiple of one row to another row
• These row operations do not change the determinant of the matrix

Slide 27

• Example 7: Find the inverse of the matrix G = [[1, 2], [3, 7]] using elementary row operations
• Step 1: Perform row operations on [G | I]
  • Multiply R1 by -3 and add it to R2 to get [1, 0 | -1, -1]
  • Multiply R2 by 2/3 to obtain the identity matrix on the left side
• Step 2: The inverse of G is [0, 1 | -2/3, 1/3]

Slide 28

• Example 8: Find the inverse of the matrix H = [[1, 2, 3], [0, 1, 4], [5, 6, 0]] using elementary row operations
• Step 1: Perform row operations on [H | I]
  • Multiply R3 by -1/5 and add it to R1 to get [1, 2/5, 3/5 | 0, 1/5, -4/5]
  • Multiply R2 by -6 and add it to R3 to get [1, 2/5, 3/5 | 0, 1/5, -4/5 | -1, 0, 0]
• Step 2: The inverse of H is [1, 2/5, 3/5 | 0, 1/5, -4/5 | -1, 0, 0]

Slide 29

• The inverse of a matrix can be used to solve systems of linear equations
• It involves multiplying the inverse matrix with the matrix of variables to obtain the solution matrix
• Example:
  • Consider the system of equations:
    • 2x + 3y - z = 1
    • 4x - y + 2z = 2
    • x + y + z = 3
  • The coefficient matrix A = [[2, 3, -1], [4, -1, 2], [1, 1, 1]]
  • The variable matrix X = [[x], [y], [z]]
  • The constant matrix B = [[1], [2], [3]]
  • Multiply both sides of the equation AX = B by A^(-1) to obtain X

Slide 30

• Recap and summary of the topic “Determinants - Inverse of a matrix”:
  • The determinant of a matrix is a numerical value associated with it
  • The inverse of a matrix exists only if its determinant is non-zero
  • Inverse can be found using adjugate method or elementary row operations
  • Inverse matrices are useful for solving systems of linear equations efficiently
  • The inverse of a diagonal matrix is obtained by taking the reciprocal of each diagonal element
  • Elementary row operations can also be used to find the inverse of a matrix.