Matrices - Examples

• What are Matrices?
• How to represent Matrices?
• Different Types of Matrices
  • Square Matrix
  • Row Matrix
  • Column Matrix
  • Zero Matrix
  • Identity Matrix
• Addition and Subtraction of Matrices
• Scalar Multiplication of Matrices
• Multiplication of Matrices
• Properties of Matrix Operations
• Example Problems

Slide 11

• Addition and Subtraction of Matrices
  • Matrices can be added or subtracted only if they have the same dimensions.
  • To add or subtract matrices, simply add or subtract corresponding elements.
  • Example:
    • A = [3 4 -2; 0 2 1; -1 5 7]
    • B = [1 0 3; 2 -1 2; 4 6 0]
    • A + B = [4 4 1; 2 1 3; 3 11 7]
    • A - B = [2 4 -5; -2 3 -1; -5 -1 7]

Slide 12

• Scalar Multiplication of Matrices
  • A scalar can be multiplied to a matrix by multiplying each element of the matrix by the scalar.
  • Example:
    • A = [2 -4 5; 3 0 -1; 6 2 7]
    • 3A = [6 -12 15; 9 0 -3; 18 6 21]

Slide 13

• Multiplication of Matrices
  • The multiplication of two matrices can be performed if the number of columns in the first matrix is equal to the number of rows in the second matrix.
  • The resulting matrix will have the number of rows from the first matrix and the number of columns from the second matrix.
  • Example:
    • A = [1 2; 3 4; 5 6]
    • B = [2 4 6; 1 3 5]
    • AB = [4 10 16; 10 26 42; 16 42 68]

Slide 14

• Properties of Matrix Operations
  • Matrix addition is commutative: A + B = B + A
  • Matrix addition is associative: (A + B) + C = A + (B + C)
  • Scalar multiplication is associative: (ab)A = a(bA)
  • Scalar multiplication distributes over matrix addition: a(A + B) = aA + aB
  • Matrix multiplication is distributive over matrix addition: A(B + C) = AB + AC

Slide 15

• Example Problem 1:
  • A = [2 3; -1 4]
  • B = [5 1; 2 -3]
  • C = [3 0; 1 2]
  • Calculate: 2A - B + 3C

Slide 16

• Example Problem 2:
  • A = [1 2 -3; 4 5 6]
  • B = [2 0 1]
  • C = [3; -1; 2]
  • Calculate: AB + BA - C

Slide 17

• Example Problem 3:
  • A = [2 -1 3; 0 4 1]
  • B = [1 2; -1 3; 5 0]
  • Calculate: AB

Slide 18

• Example Problem 4:
  • A = [4; 2]
  • B = [3 -1]
  • Calculate: A^T x B

Slide 19

• Example Problem 5:
  • A = [2 -1]
  • Calculate: A x A^T

Slide 20

• Summary and Conclusion
  • Matrices are mathematical structures that consist of rows and columns of numbers.
  • Matrices can be added, subtracted, multiplied by scalars, and multiplied together.
  • Properties of matrix operations such as commutativity and distributivity hold true.
  • Matrix operations are widely used in various areas of mathematics, engineering, and computer science.

Slide 21

• Example Problem 6:
  • A = [2 -1; 3 0]
  • B = [1 2; -1 3]
  • Calculate: (A + B)^T
• Example Problem 7:
  • A = [2 -1; 3 0]
  • B = [1 2; -1 3]
  • Calculate: A^T + B^T
• Example Problem 8:
  • A = [2 -1; 3 0]
  • B = [1; -1]
  • Calculate: AB
• Example Problem 9:
  • A = [1 2 3; 4 5 6; 7 8 9]
  • B = [2 -1 3; 0 4 1; -2 0 5]
  • Calculate: A^T B^T
• Example Problem 10:
  • A = [1 2; -1 3]
  • B = [2 -1; 0 4]
  • Calculate: (A - B)^T

Slide 22

• Example Problem 11:
  • A = [2 -1 3; 0 4 1]
  • B = [4 2; -1 6; 3 0]
  • Calculate: AB
• Example Problem 12:
  • A = [1 2 3; 4 5 6]
  • B = [2 -1; 0 4; 3 0]
  • Calculate: BA
• Example Problem 13:
  • A = [2 -1; 3 0]
  • B = [1 -3; 2 5]
  • Calculate: AB
• Example Problem 14:
  • A = [2 -1; 3 0]
  • B = [1 -3; 2 5]
  • Calculate: BA
• Example Problem 15:
  • A = [1 2; 3 4]
  • B = [1 2; 3 4]
  • Calculate: A^2

Slide 23

• Example Problem 16:
  • A = [2 -1 3; 0 4 1; -2 0 5]
  • Calculate: A^2
• Example Problem 17:
  • A = [2 -1 3; 0 4 1; -2 0 5]
  • Calculate: A^3
• Example Problem 18:
  • A = [1 2; 3 4]
  • Calculate: A^3
• Example Problem 19:
  • A = [2 -1; 3 0]
  • Calculate: det(A)
• Example Problem 20:
  • A = [2 -1 3; 0 4 1; -2 0 5]
  • Calculate: det(A)

Slide 24

• Example Problem 21:
  • A = [2 -1; 3 0]
  • Calculate: adj(A)
• Example Problem 22:
  • A = [2 -1; 3 0]
  • Calculate: inv(A)
• Example Problem 23:
  • A = [2 -1; 3 0]
  • Calculate: A^T A
• Example Problem 24:
  • A = [2 -1; 3 0]
  • Calculate: AA^T
• Example Problem 25:
  • A = [-2 1; -3 0]
  • Calculate: adj(A)

Slide 25

• Example Problem 26:
  • A = [-2 1; -3 0]
  • Calculate: inv(A)
• Example Problem 27:
  • A = [-2 1; -3 0]
  • Calculate: A^T A
• Example Problem 28:
  • A = [-2 1; -3 0]
  • Calculate: AA^T
• Example Problem 29:
  • A = [2 -1; -3 4]
  • Calculate: adj(A)
• Example Problem 30:
  • A = [2 -1; -3 4]
  • Calculate: inv(A)

Slide 26

• Example Problem 31:
  • A = [2 -1; -3 4]
  • Calculate: A^T A
• Example Problem 32:
  • A = [2 -1; -3 4]
  • Calculate: AA^T
• Example Problem 33:
  • A = [1 2; 3 4]
  • Calculate: adj(A)
• Example Problem 34:
  • A = [1 2; 3 4]
  • Calculate: inv(A)
• Example Problem 35:
  • A = [1 2; 3 4]
  • Calculate: A^T A

Slide 27

• Example Problem 36:
  • A = [1 2; 3 4]
  • Calculate: AA^T
• Example Problem 37:
  • A = [2 -1; 3 0]
  • Calculate: eigenvalues of A
• Example Problem 38:
  • A = [2 -1; 3 0]
  • Calculate: eigenvectors of A
• Example Problem 39:
  • A = [1 2; 3 4]
  • Calculate: eigenvalues of A
• Example Problem 40:
  • A = [1 2; 3 4]
  • Calculate: eigenvectors of A

Slide 28

• Example Problem 41:
  • A = [2 -1; -3 4]
  • Calculate: eigenvalues of A
• Example Problem 42:
  • A = [2 -1; -3 4]
  • Calculate: eigenvectors of A
• Example Problem 43:
  • A = [-2 1; -3 0]
  • Calculate: eigenvalues of A
• Example Problem 44:
  • A = [-2 1; -3 0]
  • Calculate: eigenvectors of A
• Example Problem 45:
  • A = [1 -2; 3 -4]
  • Calculate: eigenvalues of A

Slide 29

• Example Problem 46:
  • A = [1 -2; 3 -4]
  • Calculate: eigenvectors of A
• Example Problem 47:
  • A = [-1 2; -3 4]
  • Calculate: eigenvalues of A
• Example Problem 48:
  • A = [-1 2; -3 4]
  • Calculate: eigenvectors of A
• Example Problem 49:
  • A = [0 2; -3 0]
  • Calculate: eigenvalues of A
• Example Problem 50:
  • A = [0 2; -3 0]
  • Calculate: eigenvectors of A

Slide 30

• Summary and Conclusion
  • Matrices are important mathematical structures used in various fields such as mathematics, physics, engineering, and computer science.
  • Matrix operations such as addition, subtraction, scalar multiplication, and matrix multiplication are fundamental operations performed on matrices.
  • Properties of matrix operations such as commutativity, distributivity, and associativity hold true.
  • Determinant, adjoint, inverse, and eigenvalues/vectors are important concepts related to matrices.
  • Understanding matrices and their operations is crucial for solving complex mathematical problems and real-world applications.