Higher Secondary Mathematics: 19.8 - Properties of Multiplication of Matrices

In the previous section, we saw that, multiplication of matrices is not commutative. In this section, we will see some properties of multiplication of matrices.

Multiplication of matrices possesses three properties.
Property I: The associative law
• If A, B and C are three matrices, then (AB)C = A(BC)
• For this property to be applicable, the following matrices should be defined:
AB, BC, A(BC) and (AB)C.
• Let us see an example. It can be written in 4 steps:

1. Let the three matrices be:
A = $\left[\begin{array}{r}
7    &{    -4    }    &{    3    }    \\
-2    &{    6    }    &{    8    }    \\
-3    &{    9    }    &{    -2    }    \\
\end{array}\right]
$, B = $\left[\begin{array}{r}
2    &{    1    }    \\
-4    &{    6    }    \\
5    &{    3    }    \\
\end{array}\right]
$ and C = $\left[\begin{array}{r}
3    &{    5    }    &{    9    }    &{    0    }    \\
6    &{    7    }    &{    4    }    &{    2    }    \\
\end{array}\right]
$

2. First we find (AB)C:

3. Next we find A(BC):

4. Based on (2) and (3), we can write:
(AB)C = A(BC)

◼ We will see the actual proof in higher classes.

Property II: The distributive law
• If A, B and C are three matrices, then:
(i) A(B+C) = AB + AC
(ii) (A+B)C = AC + BC
• For this property to be applicable, the following matrices should be defined:
AB, BC, AC, A(B+C) and (A+B)C.

• Let us see an example for part (i). It can be written in 4 steps:

1. Let the three matrices be:
A = $\left[\begin{array}{r}
10    &{    -7    }    &{    14    }    \\
7    &{    31    }    &{    3    }    \\
\end{array}\right]$,
B = $\left[\begin{array}{r}
-7    &{    0    }    \\
8    &{    -2    }    \\
6    &{    3    }    \\
\end{array}\right]$
and C = $\left[\begin{array}{r}
3   &{   -4   }   \\
12   &{   15   }   \\
9   &{   8   }   \\
\end{array}\right]$

2. First we find A(B+C):

3. Next we find AB + AC:

4. Based on (2) and (3), we can write:
A(B+C) = AB+AC

◼ We will see the actual proof in higher classes.

• Let us see an example for part (ii). It can be written in 4 steps:

1. Let the three matrices be:
A = $\left[\begin{array}{r}
-8    &{    2    }    &{    5    }    \\
0    &{    -7    }    &{    -3    }    \\
3    &{    2    }    &{    4    }    \\
\end{array}\right]$,
B = $\left[\begin{array}{r}
3   &{   9   }   &{   12   }   \\
7   &{   -4   }   &{   5   }   \\
8   &{   6   }   &{   15   }   \\
\end{array}\right]$
and C = $\left[\begin{array}{r}
6     \\
4    \\
-7     \\
\end{array}\right]
$

2. First we find (A+B)C: