Laws of matrix algebra

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We can add and multiply matrices. These operations may or may not satisfy certain laws or rules that characterize matrix algebra. We also introduce an algebraic structure called a ring and show that the set of square matrices is a ring.



In what follows, we assume that matrices \(A, B\) and \(C\) are such that their shapes are compatible with matrix addition and multiplication.

Then you should be able to show at least the first two (distributive laws) of the following

\[\begin{eqnarray} A(B + C) &=& (AB) + (AC),\\ (A + B)C &=& (AC) + (BC),\\ (AB)C &=& A(BC). \end{eqnarray} \]

Let us show the last one.

Theorem (Associativity of matrix multiplication)

Let \(A = (a_{ij}), B=(b_{ij})\) and \(C=(c_{ij})\) be matrices. Suppose that \(A\) has \(r\) columns and \(B\) has \(r\) rows, and that \(B\) has \(s\) columns and \(C\) has \(s\) rows. Then \(A(BC) = (AB)C\). In other words, matrix multiplication is associative.

Proof. Let \(X = (x_{ij}) = A(BC)\) and \(Y = (y_{ij}) = (AB)C\). Then

\[x_{ij} = \sum_{k=1}^{r}a_{ik}\left(\sum_{l=1}^{s}b_{kl}c_{lj}\right)\]

and

\[y_{ij} = \sum_{l=1}^{s}\left(\sum_{k=1}^{r}a_{ik}b_{kl}\right)c_{lj}.\]

And these are both equal to the same sum:

\[\sum_{k=1}^{r}\sum_{l=1}^{s}a_{ik}b_{kl}c_{lj}.\]

■

Let \(M_n(K)\), or simply \(M_n\) when there is no confusion, denote the set of all \(n\times n\) matrices over the field \(K\) (i.e., matrices with elements of \(K\)). Then \(M_n(K)\) is closed under matrix addition and multiplication. As usual, we assume \(K = \mathbb{R}\) and \(M_n = M_n(\mathbb{R})\) in the following unless otherwise stated.

Let \(O = (o_{ij})\in M_n\) be the ``zero'' matrix where all elements are 0. For any \(A \in M_n\), we have

\[A + O = O + A = A.\]

For any \(A = (a_{ij}) \in M_n\), we define \(-A = (-a_{ij})\). Clearly, \(-A \in M_n\), and 

\[A + (-A) = O.\] 

That is, the additive inverse of a matrix always exists. 

Let us summarize the laws of matrix algebra. In the following, \(A, B, C\in M_n\), and ``addition'' and ``multiplication'' mean matrix addition and matrix multiplication, respectively.

  1. \(A + B\in M_n.\) (\(M_n\) is closed under addition.)
  2. \((A + B) + C = A + (B+C).\) (Addition is associative.)
  3. The additive identity (the zero matrix) \(O\in M_n\) exists such that \(A + O = O + A = A\) holds for any \(A \in M_n.\)
  4. For any \(A\in M_n\), there exists its additive inverse \(-A \in M_n\) such that \(A + (-A) = O.\)
  5. \(A + B = B + A.\) (Addition is commutative.)
  6. \(AB\in M_n.\) (\(M_n\) is closed under multiplication.)
  7. \((AB)C = A(BC).\) (Multiplication is associative.)
  8. \(A(B+C) = AB + AC\). (Distributive law.)
  9. \((A+B)C = AC + BC.\) (Distributive law.)
Actually, it is not only \(M_n\). There are many sets equipped with addition and multiplication that satisfy these laws. They are so common that we have a special name for them.

Definition (Ring)

A set endowed with ``addition'' and ``multiplication'' that satisfies the above nine axioms is called a ring.

Thus, \(M_n\) is a ring with matrix addition and matrix multiplication.

Exercise. Compare the ring axioms with the field axioms. □
See also: Fields. 

Example. \(\mathbb{Z}\), with usual addition and multiplication, is a ring. □

Example. \(\mathbb{Q}[x]\), the set of polynomials in \(x\) with rational coefficients, is a ring. (What are addition and multiplication?) □


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