Math

• What is a matrix?


  A rectangular array of numbers
• How are matrices typically expressed?

  In brackets, either ( ) or [ ]
• What does the term 'm by n' refer to in matrices?

  It refers to the number of rows and columns
• What is the definition of the ijth element of a matrix?

  Element in the ith row and jth column
• What is the notation for a matrix?

  A = ( ) or A = [ ]
• What is the size or order of a matrix?

  It is denoted as m×n
• If matrix A has 3 rows and 4 columns, what is its order?

  3×4

  
• What is a column matrix?

  A matrix with only one column
• What is a row matrix?

  A matrix with only one row
• What is a square matrix?

  A matrix with the same number of rows and columns
• What is a zero/null matrix?

  A matrix with all entries 0
• When are two matrices said to be equal?

  When their corresponding elements are equal
• How do you find the sum of two matrices A and B?

  Add their corresponding elements
• How do you find the difference of two matrices A and B?

  Subtract their corresponding elements
• What is the commutative property of matrix addition?

  A + B = B + A
• What is the associative property of matrix addition?

  (A + B) + C = A + (B + C)
• What is the additive identity in matrix addition?

  A + 0 = A
• What is the additive inverse in matrix addition?

  A + (−A) = 0
• What does scalar multiplication of a matrix involve?

  Multiplying each element by a scalar
• If k is a scalar and A is a matrix, what is kA?

  Matrix obtained by multiplying A by k
• What is the condition for the product of two matrices A and B to be defined?

  Columns of A must equal rows of B
• What is the order of the product of matrices A and B?

  m by n if A is m by p and B is p by n
• What is the associative property of matrix multiplication?

  A(BC) = (AB)C
• What is the left distributive property of matrix multiplication?

  A(B + C) = AB + AC
• What is the right distributive property of matrix multiplication?

  (B + C)A = BA + CA
• What is the identity matrix?

  A square matrix with diagonal elements 1
• What is the trace of a matrix?

  The sum of diagonal elements
• What is the transpose of a matrix?

  Matrix obtained by interchanging rows and columns
• What is a symmetric matrix?

  A matrix where At = A
• What is a skew-symmetric matrix?

  A matrix where At = -A
• What happens to the diagonal elements of a skew-symmetric matrix?

  All main diagonal elements are zero
• What are the properties of scalar multiplication of matrices?

  • r(A + B) = rA + rB
  • (r + s)A = rA + sA
  • (rs)A = r(sA)
  • 1A = A
  • 0A = 0
• What are the properties of matrix multiplication?

  1. A(BC) = (AB)C (Associative property)
  2. A(B + C) = AB + AC (Left Distributive property)
  3. (B + C)A = BA + CA (Right Distributive property)
  4. k(AB) = (kA)B = A(kB)
  5. 0A = 0 = A0
• What are the conditions for the existence of matrix operations?

  1. A + B: Same order
  2. AB: Columns of A = Rows of B
  3. BA: Rows of B = Columns of A
  4. AB + BA: Same order

  
• How can a matrix be expressed as the sum of symmetric and skew-symmetric matrices?

  • A = (A + At)/2 (Symmetric part)
  • A = (A - At)/2 (Skew-symmetric part)
• What are the types of matrices discussed?

  • Column matrix
  • Row matrix
  • Square matrix
  • Zero/null matrix
• What are the key definitions related to matrices?

  • Matrix: Rectangular array of numbers
  • Order: m×n
  • Equal matrices: Corresponding elements are equal
  • Transpose: Interchanging rows and columns
• What is the significance of the identity matrix in multiplication?

  • AI_n = A = I_nA for any square matrix A
• What is the zero product rule in matrix multiplication?

  • ab=0 implies a=0 or b=0 or both

  
• What is the cancellation law in matrix multiplication?

  • ab=ac implies b=c, not applicable for matrices