Finding Single Entries & Multiplying by the Identity Matrix

However, you should expect to have to do at least a few multiplications by hand, maybe on the "no calculator" portion of the test, just to show that you *can* actually do it.

One special type of question for multiplying matrices is where you're asked to find exactly one entry of the matrix product. That is, rather than wanting you to do the entire multiplication, they're wanting you to do the multiplication and addition that results in one particular entry of the product.

And it is possible to create questions that require that you be able do find that exactly one entry — where you cannot get help from your calculator — so make sure that you can do this.

How can you find just a specific entry in a matrix multiplication?

In order to find a particular entry a_i,j in a matrix multiplication, multiply the i-th row of the left-hand matrix by the j-th column of the right-hand matrix.

• Given the following matrices A and B, and defining C as AB = C, find the values of entries c_3,2 and c_2,3 in matrix C.
   
  

The above type of exercise serves as a reminder that, in general, to find the entry c_i,j of a product matrix, you multiply row i of A against column j of B.

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What is the identity matrix?

The identity matrix I is a square matrix containing all zeroes, except for 1's on the top-left to bottom-right diagonal. Multiplying a matrix by the identity matrix is much like multiplying a number by the number 1, in that nothing changes as a result of the multiplication.

But to say "the" identity matrix is misleading; identity matrices come in different sizes, just as do regular matrices. The identity matrix with respect to a given matrix is the square matrix whose dimensions make the matrix multiplication for that matrix defined. There is more than just one multiplicative identity for matrices because the particular identity you need till depend upon the size of the matrix against which it is being multiplied.

Because multiplying a matrix by the identity matrix I doesn't change anything — just like multiplying a number by 1 doesn't change anything — each of the matrix identity I and the regular number 1 are called the "multiplicative identity" for their respective areas of influence.

For example, suppose you have the following matrix A:

To multiply A on the right by the identity (that is, to do AI), you have to use I₃, the 3×3 identity, in order to have the right number of rows for the multiplication to work:

On the other hand, to multiply A on the left by the identity, you have to use I₂, the 2×2 identity, in order to have the right number of columns:

As you can see, if you are dealing with a non-square matrix such as the matrix A in the above example, the identity matrix that you use will depend upon the side of A on which you're multiplying.

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.

What is the relationship between matrix multiplication and the Commutative Property?

There is no relationship between matrix multiplication and the Commutative Property, other than that matrix multiplication is not commutative — that is, you can not switch around the factors in matrix multiplication, willy-nilly, and expect to get the same result.

Matrix multiplication is probably the first time that the Commutative Property has ever been an issue for you. You may remember when they made a big deal, starting back in middle school or so, about how ab = ba or 5×6 = 6×5? This rule or property probably seemed fairly stupid at the time, because you already knew that order didn't matter in multiplication. So what was their point, exactly?

Introducing you to those rules back then was actually probably kind of pointless, since order didn't matter for anything you were multiplying then. Well, now the Commutative Property *does* matter, because order *does* matter for matrix multiplication.

Always keep in mind that, for matrices, AB almost never equals BA. There will even be times when multiplying in the other order won't even be defined.