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Matrix Multiplication / The Identity Matrix (page 3 of 3)


Here are a couple more examples of matrix multiplication:

  • Find CD and DC, if they exist, given that C and D are the following matrices:

    C = [[ 2  -1 ][ 0  3 ][ 1  0 ]] ;  D = [[ 0  1  4  1 ][ -2  0  0  2 ]]

    C is a 32 matrix and D is a 24 matrix, so first I'll look at the dimension product for CD:

      (32)(24): inner dimensions match, so product is defined; outer dimensions indicate a 34 product matrix

    So the product CD is defined (that is, I can do the multiplication); also, I can tell that I'm going to get a 34 matrix for my answer. Here's the multiplication:

      Shows the computations involved in multiplying the two matrices.

    However, look at the dimension product for DC:

      (24)(32): the inner dimensions do not match, so multiplication is not possible

    Since the inner dimensions don't match, I can't do the multiplication. (The columns of C aren't the same length as the rows of D; the columns of C are too short, or, if you prefer, the rows of D are too long.) Then the answer is:

      CD = [[ 2  2  8  -4 ][ -6  0  0  6 ][ 0  1  4  -1 ]]

      DC is not defined.

  • Given the following matrices A and B, and defining C as AB = C, find the values of entries c3,2 and c2,3 in matrix C.
    • A = [[ 4  3  6  -1 ][ 0  2  1  4 ][ 3  -2  -2  -2 ][ 0  0  1  3 ]] ;  B = [[ 2  3  0 ][ 0  4  -2 ][ 1  0  -2 ][ 0  -1  0 ]]

    The dimension product of AB is (44)(43), so the multiplication will work, and C will be a 43 matrix. But to find c3,2, I don't need to do the whole matrix multiplication. The 3,2-entry is the result of multiplying the third row of A against the second column of B, so I'll just do that:

      c3,2 = (3)(3) + (2)(4) + (2)(0) + (2)(1) = 9 8 + 0 + 2 = 3

    On the other hand, c2,3 is the result of multiplying the second row of A against the third column of B, so:   Copyright Elizabeth Stapel 2003-2011 All Rights Reserved

      c2,3 = (0)(0) + (2)(2) + (1)(2) + (4)(0) = 0 4 2 + 0 = 6

      c3,2 = 3 and c2,3= 6

This type of problem serves as a reminder that, in general, to find ci,j you multiply row i of A against column j of B.


Multiplying by the identity

Multiplying by the identity matrix I doesn't change anything, just like multiplying a number by 1 doesn't change anything. This property is why I and 1 are each called the "multiplicative identity". But while there is only one "multiplicative identity" for regular numbers (namely the number 1), there are lots of different identity matrices. Why? Because the identity matrix you need will depend upon the size of the matrix that it is being multiplied on! For instance, suppose you have the following matrix A:

    A = [[ 1  2  3 ][ 4  5  6 ]]

To multiply A on the right by the identity (that is, to do AI ), you have to use I3, the 33 identity, in order to have the right number of rows for the multiplication to work:

    Displays the multiplication of A by I_3

On the other hand, to multiply A on the left by the identity, you have to use I2, the 22 identity, in order to have the right number of columns:

    Displays the multiplication of I_2 by A

That is, if you are dealing with a non-square matrix (such as A in the above example), the identity matrix you use will depend upon the side that you're multiplying on. This is just another example of matrix weirdness. Don't let it scare you. Matrices aren't bad; they're just different... really, really different.

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Cite this article as:

Stapel, Elizabeth. "Matrix Multiplication / The Identity Matrix." Purplemath. Available from
    http://www.purplemath.com/modules/mtrxmult3.htm. Accessed
 

 



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