19.9 - Multiplicative Identity

In the previous section, we saw two properties of multiplication of matrices. In this section, we will see the third property.

Property III: The existence of multiplicative identity
• Consider any square matrix A of the order (m×m).
• We can write an identity matrix I of the same order.
That I will satisfy the equation: AI = IA = A
• Let us see an example. It can be written in 4 steps:

1. Let A = $\left[\begin{array}{r}                           
-8    &{    2    }    &{    5    }    \\
0    &{    -7    }    &{    -3    }    \\
3    &{    2    }    &{    4    }    \\
\end{array}\right]$

• Then I = $\left[\begin{array}{r}                           
1    &{    0    }    &{    0    }    \\
0    &{    1    }    &{    0    }    \\
0    &{    0    }    &{    1    }    \\
\end{array}\right]                           
$               

2. First we find AI:

3. Next we find IA:

4. Based on (2) and (3), we can write:
AI = IA = A

◼ We will see the actual proof in higher classes.

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Now we will see a solved example.

Solved example 19.14
If A = $\left[\begin{array}{r}                           
1    &{    3    }    &{    2    }    \\
2    &{    0    }    &{    -1    }    \\
1    &{    2    }    &{    3    }    \\
\end{array}\right]                           
$, then show that A³ - 4A² - 3A + 11I = O
Solution:
1. First we will write A²:

2. Next we will write A³:

3. Next we will write 4A²:

3. Next we will write 3A:

4. Finally we will write 11I:

5. Substituting the values, we get:

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The link below gives a few more solved examples:

Exercise 19.2

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In the next section, we will see transpose of a matrix.

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