Algebraic Properties of Matrix Operations

In this page, we give some general results about the three operations: addition, multiplication, and multiplication with numbers, called scalar multiplication.

From now on, we will not write (mxn) but mxn.

Properties involving Addition. Let A, B, and C be mxn matrices. We have

1.
A+B = B+A
2.
(A+B)+C = A + (B+C)
3.
$A + {\cal O} = A$
where $\cal O$ is the mxn zero-matrix (all its entries are equal to 0);
4.
$A+B = {\cal O}$ if and only if B = -A.

Properties involving Multiplication.

1.
Let A, B, and C be three matrices. If you can perform the products AB, (AB)C, BC, and A(BC), then we have

(AB)C = A (BC)

Note, for example, that if A is 2x3, B is 3x3, and C is 3x1, then the above products are possible (in this case, (AB)C is 2x1 matrix).
2.
If $\alpha$ and $\beta$ are numbers, and A is a matrix, then we have

\begin{displaymath}\alpha (\beta A) = (\alpha \beta) A\end{displaymath}

3.
If $\alpha$ is a number, and A and B are two matrices such that the product $A\cdot B$ is possible, then we have

\begin{displaymath}\alpha (AB) = (\alpha A)B = A (\alpha B)\end{displaymath}

4.
If A is an nxm matrix and $\cal O$ the mxk zero-matrix, then

\begin{displaymath}A {\cal O} = {\cal O}\end{displaymath}

Note that $A {\cal O}$ is the nxk zero-matrix. So if n is different from m, the two zero-matrices are different.

Properties involving Addition and Multiplication.

1.
Let A, B, and C be three matrices. If you can perform the appropriate products, then we have

(A+B)C = AC + BC

and

A(B+C) = AB + AC

2.
If $\alpha$ and $\beta$ are numbers, A and B are matrices, then we have

\begin{displaymath}\alpha (A+B) = \alpha A + \alpha B\end{displaymath}

and

\begin{displaymath}(\alpha +\beta)A = \alpha A + \beta B\end{displaymath}

Example. Consider the matrices

\begin{displaymath}A = \left(\begin{array}{cc}
0&1\\
-1&0\\
\end{array}\right)...
...nd}\; C = \left(\begin{array}{ccc}
0&1&5\\
\end{array}\right).\end{displaymath}

Evaluate (AB)C and A(BC). Check that you get the same matrix.

Answer. We have

\begin{displaymath}AB = \left(\begin{array}{c}
-1\\
-2\\
\end{array}\right)\end{displaymath}

so

\begin{displaymath}(AB)C = \left(\begin{array}{c}
-1\\
-2\\
\end{array}\right)...
...t(\begin{array}{ccc}
0&-1&-5\\
0&-2&-10\\
\end{array}\right).\end{displaymath}

On the other hand, we have

\begin{displaymath}BC = \left(\begin{array}{ccc}
0&2&10\\
0&-1&-5\\
\end{array}\right)\end{displaymath}

so

\begin{displaymath}A(BC) = \left(\begin{array}{cc}
0&1\\
-1&0\\
\end{array}\ri...
...t(\begin{array}{ccc}
0&-1&-5\\
0&-2&-10\\
\end{array}\right).\end{displaymath}

Example. Consider the matrices

\begin{displaymath}X = \left(\begin{array}{c}
a\\
b\\
c\\
\end{array}\right)\...
...ray}{cccc}
\alpha & \beta & \nu & \gamma\\
\end{array}\right).\end{displaymath}

It is easy to check that

\begin{displaymath}X = a \left(\begin{array}{c}
1\\
0\\
0\\
\end{array}\right...
...) + c \left(\begin{array}{c}
0\\
0\\
1\\
\end{array}\right) \end{displaymath}

and

\begin{displaymath}Y = \alpha \left(\begin{array}{cccc}
1 & 0 & 0 & 0\\
\end{ar...
...ma \left(\begin{array}{cccc}
0 &0 & 0& 1\\
\end{array}\right).\end{displaymath}

These two formulas are called linear combinations. More on linear combinations will be discussed on a different page.

We have seen that matrix multiplication is different from normal multiplication (between numbers). Are there some similarities? For example, is there a matrix which plays a similar role as the number 1? The answer is yes. Indeed, consider the nxn matrix

\begin{displaymath}I_n = \left(\begin{array}{ccccc}
1&0&0&\cdots&0\\
0&1&0&\cdo...
...cdot\\
\cdot&&&&\cdot\\
0&0&0&\cdots&1\\
\end{array}\right).\end{displaymath}

In particular, we have

\begin{displaymath}I_2 = \left(\begin{array}{ccc}
1&0\\
0&1\\
\end{array}\righ...
...egin{array}{ccc}
1&0&0\\
0&1&0\\
0&0&1\\
\end{array}\right).\end{displaymath}

The matrix In has similar behavior as the number 1. Indeed, for any nxn matrix A, we have

A In = In A = A

The matrix In is called the Identity Matrix of order n.

Example. Consider the matrices

\begin{displaymath}A = \left(\begin{array}{cc}
1&2\\
-1&-1\\
\end{array}\right...
...B = \left(\begin{array}{cc}
-1&-2\\
1&1\\
\end{array}\right).\end{displaymath}

Then it is easy to check that

\begin{displaymath}AB = I_2 \;\;\mbox{and}\;\; BA = I_2.\end{displaymath}

The identity matrix behaves like the number 1 not only among the matrices of the form nxn. Indeed, for any nxm matrix A, we have

\begin{displaymath}I_n A = A\;\;\mbox{and}\;\; A I_m = A.\end{displaymath}

In particular, we have

\begin{displaymath}I_4 \left(\begin{array}{c}
a\\
b\\
c\\
d\\
\end{array}\ri...
... \left(\begin{array}{c}
a\\
b\\
c\\
d\\
\end{array}\right).\end{displaymath}

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Author: M.A. Khamsi

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