Linear Algebra Ch.2 Flash Cards

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Title: Linear Algebra Ch.2
Description: Operations with Matrices
Number of Cards: 19
Save Count: 3
Author: xyz71411
Created: 2002-10-18
Tags: college
Private No

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    • Question
    • Answer
    • Side 3
    • The ________ of a matrix is formed by writing its columna as rows.
    • transpose

      examples:
      |2|
      A =|8|, A^T =[2 8]
    • A matrix that has only one column is called a _____ matrix or ______ vector.
    • column, column

      [ 2 ]
      A = | 3 |
      [ 2 ]
    • FINDING THE INVERSE OF A MATRIX BY GAUSS-JORDAN ELIMINATION
    • Let A be a square matrix of order n.
      1. Write the n*2n matrix that consists of the given matrix A on the left and the n*n identity matrix I on the right to obtain [A : I]. Note that you separate the matrices A and I by a dotted line. This process is called adjoining the matrices A and I.
      2. If possible, row reduce A to I using ERO's on the entire matrix [A : I]. The result will be the matrix [I : A^-1]. If this is not possible, then A is not invertible.
      3. Check your work by multiplying to see that AA^-1 = I = A^-1*A.
    • Two matrices A = [aij] and B = [bij] are ________ if they have the same size (m * n) and aij = bij for 1 <= i <= m and 1 <= j <= n.
    • equal
    • Properties of Zero Matrices:
      If A is an m*n matrix and c is a scalar, then the following properties are true.
      1. A + Omn = ?
      2. A + (-A) = ?
      3. if cA = Omn,, then c = ? or A = ?
    • 1. = A
      2. = Omn
      3. = 0 or = Omn
    • A matrix that has only one row is called a ______ matrix or ______ vector.
    • row, row

      A = [2 3 4]
    • If A is an invertible matrix, then its inverse is ______. The inverse of A is denoted by A^-1.
    • unique
    • If A = [aij] and B = [bij] are matrices of size m*n, then their _____ is the m*n matrix given by

      A + B = [aij + bij]
    • sum, The sum of two matrices of different sizes is undefined.

      |-1 2| + | 1 3| = |-1+1 2+3| = | 0 5|
      | 0 1| |-1 2| | 0-1 1+2| |-1 3|
    • Find the product AB, where
      |-1 3| |-3 2|
      A = | 4 -2| and B =|-4 1|
      | 5 0|
    • |-9 1|
      |-4 6|
      |-15 10|
    • Real numbers are referred to as ______.
    • scalars
    • A matrix that does not have an inverse is called ______ (or _______).
    • noninvertible
      singular
    • You can use -A to represent the scalar product ______.
    • (-1)A

      A - B = A + (-B)
    • Properties of Matrix Multiplication
      If A, B, and C are matrices (with sizes such that the given matrix products are defined) and c is a scalar, then the following properties are true.
      1. A(BC) = ?
      2. A(B + C) = ?
      3. (A + B)C = ?
      4. c(AB) = ?
    • 1. (AB)C (associative property of multiplication)
      2. AB + AC (distributive property)
      3. AC + BC (distributive property)
      4. (cA)B = A(cB)
    • If A = [aij] is an m*n matrix and B = [bij] is an n*p matrix, then the product AB is an ________ matrix.
    • m*p

      AB = [cij] where

      cij = (Summation from k = 1 to n) aik*bkj = ai1*b1j + ai2*b2j +...+ ain*bnj
    • An n*n matrix A is ______ (or _______) if there exist an n*n matrix B such that
      AB = BA = In
      where (In) is the identity matrix of order n. The matrix B is called the (multiplicative) _______ of A.
    • invertible
      nonsingular
      inverse
    • If A = [aij] is an m*n matrix and c is a scalar, then the _________ _______ of A by c is the m*n matrix given by
      cA = [caij].
    • scalar multiple
    • Properties of the Identity Matrix
      If A is a matrix of size m*n, then the following properties are true.
      1. AIn = ?
      2. ImA = ?
    • 1. = A
      2. = A
    • Properties of Transpose
      If A and B are matrices (with sizes such that the given matrix operations are defined) and c is a scalar, then the following properties are true.
      1. (A^T)^T = ?
      2. (A + B)^T = ?
      3. (cA)^T = ?
      4. (AB)^T = ?
    • 1. A
      2. A^T + B^T (transpose of a sum)
      3. c(A^T) (transpose of a scalar multiple)
      4. B^T*A^T (transpose of a product)
    • Properties of matrix addition and scalar multiplication:
      1. A + B = ?
      2. A + (B + C) = ?
      3. (cd)A = ?
      4. 1A = ?
      5. c(A + B) = ?
      6. (c + d)A = ?
    • 1. = B + A (commutative property of addition)
      2. = (A + B) + C (associative property of addition)
      3. = c(dA)
      4. = A
      5. cA + cB (distributive property)
      6. cA + dA (distributive property)