Multiplication of Matrices

Key Questions

  • What is multiplication of matrices?

    Matrix multiplication is an operation performed upon two (or sometimes more) matrices, with the result being another matrix.

    This explanation will assume the student is familiar with the basics of matrices, such as matrix notation and vector dot products.

    There are certain rules which must be followed in the multiplication process. First, when multiplying any two matrices A_(rs) and B_(tu), where r and t are the number of rows in matrices A & B respectively and s and u the number of columns in matrices A & B respectively, if s!=t (that is, the number of rows in A does not equal the number of columns in B), the matrix multiplication cannot be carried out.

    When multiplying two matrices such as this, the resultant matrix AB will possess r rows and u columns; in other words, the same number of rows as the A matrix and the same number of columns as the B matrix.

    Each entry in the AB matrix will be calculated via the dot product of a row from the A matrix and a column from the B matrix. Renaming the AB matrix as C for ease of use, the value of any individual element c_ij can be found by taking the dot product of row i from A and column j from B.

    There is currently some difficulty in utilizing Socratic's math code to construct a matrix, so different notation must be used temporarily. Consider the 2x3 matrix A, such that a_11 = 1, a_12 = 0, a_13 = 3, a_21 = 0, a_22 = 5, a_23 = -1, as well as the 3x2 matrix B such that b_11 = 4, b_12 = 5, b_21 = 0, b_22 = -3, b_31 = -4, b_32 = 1. Then the resultant matrix AB = C is a 2x2 matrix, with
    c_11 = (a_11*b_11) + (a_12*b_21) + (a_13*b_31),
    c_12 = (a_11*b_12)+(a_12*b_22)+(a_13*b_32),
    c_21 = (a_21*b_11) + (a_22*b_21)+(a_23*b_31),
    c_22 = (a_21*b_12)+(a_22+b_22)+(a_23+b_32)

    Plugging in the respective values, we get c_11 = -8, c_12 = 8, c_21 = 4, c_22 = -16

    Psykolord1989 . · 2 · Dec 4 2014
  • How do I do multiplication of matrices?

    There is some information on Multiplication of Matrices here on Socratic.

    I think of it as a process that is easier to explain in person, but I'll do my best here.

    Let's go through an example:

    ((1, 2),(3, 4)) ((3, 5),(7, 11))

    Find the first row of the product

    Take the first row of ((1, 2),(3, 4)), and make it vertical in front of ((3, 5),(7, 11)). (We'll do the same for the second row in a minute.)

    It looks like:

    {: (1),(2) :}((3, 5),(7, 11))

    Now multiply times the first column and add to get the first number in the first row of the answer:
    {:(1 xx 3),(2 xx 7) :}={:(3),(14) :} now add to get 17

    The product starts with:
    ((17,"-"),("-","-"))

    Next multiply times the second column and add to get the second number in the first row of the answer:
    {:(1 xx 5),(2 xx 11) :}={:(5),(22) :} now add to get 27

    The first row of the product is: ((17,27))

    A this point we know that the product looks like:

    ((1, 2),(3, 4)) ((3, 5),(7, 11)) = ((17,27),("-","-"))

    Find the second row of the product
    Find the second row of the product by the same process using the second row of ((1, 2),(3, 4))

    {: (3),(4) :} ((3, 5),(7, 11)) to get: 9+28 = 37 and 15+44 = 59

    The second row of the product is: ((37,59))

    Write the answer

    ((1, 2),(3, 4)) ((3, 5),(7, 11)) = ((17,27),(37,59))

    Jim H · 2 · Sep 9 2015
  • What is scalar multiplication of matrices?

    Simply the multiplication of a scalar (generally a real number) by a matrix.

    The multiplication of a matriz M of entries m_(ij) by a scalar a is defined as the matrix of entries a m_(ij) and is denoted aM.

    Example:

    Take the matrix

    A=((3,14),(-4,2))

    and the scalar b=4

    Then, the product bA of the scalar b and the matrix A is the matrix

    bA=((12,56),(-16,8))

    This operation has very simple properties that are analogous to that of the real numbers.

    Vinicius M. G. Silveira · 1 · Dec 10 2014

Questions

Matrix Algebra