Blogs, Linear Algebra

Properties of Matrix Operations

Now that we have defined the matrix operations we will be using in the last post, we can now look at some properties of these operations. When looking at these operations, we often want to compare them to properties held by the real numbers. While not all of the same, many are. Here we will look at two properties of real numbers and prove that the do or do not hold for matrices.

Associativity of Addition

Theorem: The associative property holds for matrix addition.

We will prove the given theorem the matrix addition is associative. However, before we do that, we should recall what associativity is. In this case, we assume that \(A\), \(B\) and \(C\) are matrices of the same size. We then want to show that
\begin{align*}
(A+B)+C=A+(B+C).
\end{align*}
That is, we want to show that the order of the parenthesis don’t matter when adding three matrices. As a further note, matrix addition is only defined for matrices of the same size, this is why we are restricted our matrices to that condition.

Now, to show our theorem, we are really trying to show that two things are equal. We, therefore, need to start with one side of the equation and simplify it step by step until we arrive at the other side. As we try to do this, we really will just work step by step until we can make some connections.

The first thing we will do is try to use the notation we have. As such, we will denote \(A=[a_{ij}]\), \(B=[b_{ij}]\) and \(C=[c_{ij}]\). That is, we will express the matrices as arbitrary entries in the matrix. We now have
\begin{align*}
(A + B ) + C & = ([a_{ij}]+[b_{ij}])+[c_{ij}].
\end{align*}
Now, since addition of matrices is defined component wise, we know that the elements of \(A+B\) will just be the sum of the entries in \(A\) and \(B\), this is
\begin{align*}
(A + B ) + C & = ([a_{ij}]+[b_{ij}])+[c_{ij}]\\
&=[a_{ij}+b_{ij}]+[c_{ij}].
\end{align*}
That is, we wrote \(A+B\) in terms of its entries. We can do this again, combing \(C\) and we get
\begin{align*}
(A + B ) + C & = ([a_{ij}]+[b_{ij}])+[c_{ij}]\\
&=[(a_{ij}+b_{ij})+c_{ij}].
\end{align*}
Now that we’ve combined these into one matrix, we can look at the entries of this matrix. The entry in the ith row and jth column is now \((a_{ij}+b_{ij})+c_{ij}\). However, these are just real numbers. Since addition for real numbers is associative, we can rearrange these parenthesis and get
\begin{align*}
(A + B ) + C & = ([a_{ij}]+[b_{ij}])+[c_{ij}]\\
&=[(a_{ij}+b_{ij})+c_{ij}]\\
&=[a_{ij}+(b_{ij}+c_{ij})].
\end{align*}
We can now work backwards from before and break one matrix into two. Here we have,
\begin{align*}
(A + B ) + C & = ([a_{ij}]+[b_{ij}])+[c_{ij}]\\
&=[(a_{ij}+b_{ij})+c_{ij}]\\
&=[a_{ij}]+[b_{ij}+c_{ij}].
\end{align*}
Using this same step again, we finally arrive at
\begin{align*}
(A + B ) + C & = ([a_{ij}]+[b_{ij}])+[c_{ij}]\\
&=[(a_{ij}+b_{ij})+c_{ij}]\\
&=[a_{ij}]+[b_{ij}+c_{ij}] \\
&=[a_{ij}]+([b_{ij}]+[c_{ij}]) \\
&=A+(B+C).
\end{align*}
This is precisely what we wanted to show. If we were to give a formal proof, we would like to be more concise with the writing we used here. However, this would be a great way to work through the proof the first time you are working on it.

As an additional note, we assumed here that we were working on matrices with real entries. Later in this class, we will consider matrices with entries from other sets than just the real numbers. For this reason I will point out the associativity of matrix multiplication didn’t depend on the use of real numbers, but rather just that the set used for entries had associative addition. That is, with slight modification, our proof would work over any set of numbers with associative addition.

Commutativity of Matrix Multiplication

Theorem: Matrix multiplication is not commutative.

Note that we are stating that, unlike the real numbers, we will not always get the same answer if we switch the order that we multiply matrices. In order to provide a proof of this, we simply need to find an example where switching the order gives a different answer.

First Example

Let
\[A=\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\] and
\[B=\begin{bmatrix} 1 \\ 1 \end{bmatrix}\].

In this case, we note that
\[AB=\begin{bmatrix} 2 \\ 2 \end{bmatrix},\] whereas \(BA\) is not defined. Therefore, the changing the order can change whether or not the multiplication is even defined.

Second Example

While our last example was enough to show that commutativity does not hold, we can also give an example where the multiplication is defined both times, but still not equal. For this example, we’ll let
\[A=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\] and
\[B=\begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}\].

We now get that
\[AB=\begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix},\] whereas
\[BA=\begin{bmatrix} 23 & 34 \\ 31 & 46 \end{bmatrix}.\]

Therefore, again, we cannot change to order of multiplication.

Other properties

We won’t prove all of these here, but it is important that we remember the properties that the matrix operations have. On the one hand, using properties that are correct can save us time and work. On the other hand, if we try to use properties that are not true, we will end up getting things that don’t make sense. Below, we have a list of some of the properties you will be using. In each case, however, you do need to be careful as each has implication about the size of the respective matrices.

  • Matrix Addition
    • is associative,
    • is commutative,
    • has an identity,
    • has inverses.
  • Matrix multiplication
    • is associative,
    • has an identity.

Conclusion

Going through the proof and disproof of the given properties has hopefully helped you to better understand why some things work for matrices and other things don’t. The deeper understanding this gives will help you as you work through any number of problems that will arise with matrices.

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