Slide 1 - Matrices Overview

  • Matrices are rectangular arrays of numbers, symbols, or expressions.
  • Each entry in a matrix is called an element.
  • Matrices are denoted by uppercase letters.
  • The size of a matrix is given by the number of rows and columns it has.
  • For example, a matrix with 3 rows and 4 columns is called a 3x4 matrix.

Slide 2 - Matrices - Elements and Notation

  • Each element in a matrix is denoted by the corresponding row and column index.
  • The element in the ith row and jth column of a matrix A is denoted by A[i, j].
  • For example, in a matrix A = [ [2, 3, 1], [0, -1, 4] ], A[1, 2] = -1.
  • Elements of a matrix can be numbers, variables, or expressions.

Slide 3 - Matrices - Types of Matrices

  • Square Matrix: A matrix with an equal number of rows and columns.
  • Row Matrix: A matrix with only one row.
  • Column Matrix: A matrix with only one column.
  • Zero Matrix: A matrix in which all the elements are zero.
  • Identity Matrix: A square matrix having ones on its main diagonal and zeros elsewhere.
  • Diagonal Matrix: A square matrix in which all the non-diagonal elements are zero.

Slide 4 - Matrices - Operations

  • Addition of Matrices: For matrices A and B of the same size, add corresponding elements of A and B to get the sum matrix.
  • Subtraction of Matrices: Similar to addition, subtract corresponding elements of A and B to obtain the difference matrix.
  • Scalar Multiplication: Multiply each element of a matrix by a constant.
  • Matrix Multiplication: Multiply a matrix A of size m x n with a matrix B of size n x p to get a matrix C of size m x p.

Slide 5 - Matrix Addition

  • To add two matrices A and B, the matrices must have the same dimensions.
  • Add corresponding elements of A and B to obtain the sum matrix.
  • Example: A = [1, 2, 3] and B = [4, 5, 6], the sum matrix A + B = [5, 7, 9].

Slide 6 - Matrix Subtraction

  • To subtract two matrices A and B, the matrices must have the same dimensions.
  • Subtract corresponding elements of B from A to obtain the difference matrix.
  • Example: A = [7, 3, 2] and B = [2, 1, 4], the difference matrix A - B = [5, 2, -2].

Slide 7 - Scalar Multiplication

  • To perform scalar multiplication, multiply each element of the matrix by a constant.
  • Example: A = [2, 4, -1] and k = 3. The scalar product kA = [6, 12, -3].

Slide 8 - Matrix Multiplication - Introduction

  • Matrix multiplication is not commutative, i.e., AB ≠ BA in general.
  • To multiply two matrices A and B, the number of columns in A must be equal to the number of rows in B.
  • The resulting matrix will have the same number of rows as A and the same number of columns as B.
  • Example: A = [1, 2] and B = [3, 4], the matrix product AB = [11, 16].

Slide 9 - Matrix Multiplication - Example 1

  • A = [1, 2, 3]
  • B = [4]
  • AB = [14 + 21 + 3*0] = [4 + 2 + 0] = [6]

Slide 10 - Matrix Multiplication - Example 2

  • A = [1, 2] [3, 4]
  • B = [5, 6] [7, 8]
  • AB = [15 + 27] [16 + 28] [35 + 47] [36 + 48] = [19, 22] [43, 50]

Slide 11 - Matrices: Multiplication of Matrices

  • Matrix Multiplication: Multiply a matrix A of size m x n with a matrix B of size n x p to get a matrix C of size m x p.
  • The element C[i, j] of the resulting matrix C is obtained by multiplying the ith row of A with the jth column of B and summing the products.
  • The formula for matrix multiplication is C[i, j] = A[i, 1]*B[1, j] + A[i, 2]*B[2, j] + … + A[i, n]*B[n, j].
  • Example: A = [1, 2] and B = [3, 4], the matrix product AB = [11, 16].

Slide 12 - Multiplication of Matrices - Example 1

  • A = [1, 2, 3]
  • B = [4, 5, 6]
  • C = [7, 8]
  • AB = [14 + 27 + 310] [15 + 28 + 311] [44 + 57 + 610] [45 + 58 + 611] = [56, 68] [89, 110]

Slide 13 - Multiplication of Matrices - Example 2

  • A = [3, 2] [-1, 5]
  • B = [-2, 4] [3, -7]
  • AB = [3*-2 + 23] [34 + 2*-7] [-1*-2 + 53] [-14 + 5*-7] = [0, -10] [13, -39]

Slide 14 - Multiplication of Matrices - Properties

  • Matrix multiplication is associative, i.e., (AB)C = A(BC).
  • Matrix multiplication is distributive, i.e., A(B + C) = AB + AC.
  • However, matrix multiplication is not commutative, i.e., AB ≠ BA in general.

Slide 15 - Transpose of a Matrix

  • The transpose of a matrix A is obtained by interchanging its rows and columns.
  • Denoted by A^T.
  • Example: A = [1, 2, 3], its transpose A^T = [1] [2] [3]

Slide 16 - Properties of Transpose

  • (A^T)^T = A
  • (A + B)^T = A^T + B^T
  • (kA)^T = kA^T
  • (AB)^T = B^T * A^T

Slide 17 - Symmetric Matrix

  • A square matrix A is symmetric if A = A^T.
  • In other words, the element A[i, j] is equal to the element A[j, i] for all i and j.
  • Example: A = [1, 4] [4, 5] is a symmetric matrix.

Slide 18 - Skew-Symmetric Matrix

  • A square matrix A is skew-symmetric if A^T = -A.
  • In other words, the element A[i, j] is equal to the negative of the element A[j, i] for all i and j.
  • Example: A = [0, -3, 4] [3, 0, -5] [-4, 5, 0] is a skew-symmetric matrix.

Slide 19 - Scalar Multiplication of Matrices

  • To perform scalar multiplication of matrices, multiply each element of the matrix by a constant.
  • Example: A = [1, 2, 3] [4, 5, 6] and k = 2. The scalar product kA = [2, 4, 6] [8, 10, 12]

Slide 20 - Zero and Identity Matrices

  • Zero Matrix: A matrix in which all the elements are zero.
  • Identity Matrix: A square matrix having ones on its main diagonal and zeros elsewhere.
  • Example:
    • Zero Matrix of size 2x3: [0, 0, 0] [0, 0, 0]
    • Identity Matrix of size 3x3: [1, 0, 0] [0, 1, 0] [0, 0, 1]

Slide 21 - Matrices - Multiplication of Matrices

  • To multiply two matrices A and B, the number of columns in A must be equal to the number of rows in B.
  • The resulting matrix will have the same number of rows as A and the same number of columns as B.
  • The element C[i, j] of the resulting matrix C is obtained by multiplying the ith row of A with the jth column of B and summing the products.
  • The formula for matrix multiplication is C[i, j] = A[i, 1]*B[1, j] + A[i, 2]*B[2, j] + … + A[i, n]*B[n, j].
  • Example: A = [1, 2] and B = [3, 4], the matrix product AB can be calculated as follows:
    • C[1, 1] = 13 + 24 = 11
    • C[1, 2] = 10 + 21 = 2

Slide 22 - Matrix Multiplication - Example

  • A = [1, 2, 3] [4, 5, 6]
  • B = [7, 8] [9, 10] [11, 12]
  • AB = [17 + 29 + 311, 18 + 210 + 312] [47 + 59 + 611, 48 + 510 + 612] = [58, 64] [139, 154]

Slide 23 - Matrix Multiplication - Properties

  • Matrix multiplication is associative: (AB)C = A(BC)
  • Matrix multiplication is distributive over matrix addition: A(B + C) = AB + AC
  • However, matrix multiplication is not commutative: AB ≠ BA in general.

Slide 24 - Matrix Inverse

  • A square matrix A has an inverse if A^-1 exists such that A * A^-1 = A^-1 * A = I, where I is the identity matrix.
  • Not all matrices have inverses. Only non-singular matrices (determinant ≠ 0) have inverses.
  • The inverse of a matrix A can be found using the formula: A^-1 = (1/det(A)) * adj(A), where det(A) is the determinant of A and adj(A) is the adjugate of A.

Slide 25 - Matrix Determinant

  • The determinant of a square matrix A is a scalar value denoted by det(A) or |A|.
  • The determinant of a 2x2 matrix [a, b] [c, d] is given by ad - bc.
  • The determinant of a 3x3 matrix [a, b, c] [d, e, f] [g, h, i] can be calculated using the formula: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

Slide 26 - Matrix Rank

  • The rank of a matrix refers to the maximum number of linearly independent rows or columns in the matrix.
  • It is denoted by the symbol “rank(A)”.
  • The rank of a matrix can be determined by performing row operations to obtain the row echelon form of the matrix and counting the number of non-zero rows.

Slide 27 - Matrix Rank - Example

  • A = [1, 2, 3] [4, 5, 6] [7, 8, 9]
  • Now, perform row operations to obtain the row echelon form:
    • R2 = R2 - 4R1
    • R3 = R3 - 7R1
  • The row echelon form of A becomes: [1, 2, 3] [0, -3, -6] [0, 0, 0]
  • The rank of A is 2, as there are 2 non-zero rows.

Slide 28 - Matrix Equations

  • Matrix equations are equations in which matrices are involved.
  • They can be solved by performing operations to isolate the unknown variable.
  • Example: Solve the matrix equation AX = B, where A = [2, 3] [4, 5] and B = [7, 8] [9, 10]
  • Multiply both sides of the equation by A^-1 to isolate X: X = A^-1 * B.

Slide 29 - Matrix Equations - Example

  • A = [2, 3] [4, 5]
  • A^-1 = (1/(-2 + 12)) * [5, -3] [-4, 2] = [1/10, -1/10] [-2/10, 1/10]
  • B = [7, 8] [9, 10]
  • X = A^-1 * B = [1/10 * 7 + (-1/10) * 9, 1/10 * 8 + (-1/10) * 10] [-2/10 * 7 + 1/10 * 9, -2/10 * 8 + 1/10 * 10] = [1/10, 0] [-1/10, 0]

Slide 30 - Summary

  • Matrices are rectangular arrays of numbers, symbols, or expressions.
  • Matrix operations include addition, subtraction, scalar multiplication, and multiplication.
  • Matrix multiplication is not commutative, but it is associative and distributive.
  • The transpose of a matrix is obtained by interchanging its rows and columns.
  • Square matrices can be symmetric or skew-symmetric.
  • Determinants, inverses, and ranks are important properties of matrices.
  • Matrix equations can be solved by performing operations on both sides of the equation.