19.15 - Solved Examples

In the previous section, we saw a method to find the inverse of a given matrix. We saw a solved example also. In this section, we will see a few more solved examples.

Let us see some solved examples:

Solved Example 19.17
By using elementary operations, find the inverse of the matrix A = $\left[\begin{array}{r}               
0    &{    1    } &{    2    }    \\
1    &{    2    }  &{    3    }  \\
3    &{    1    }  &{    1    }  \\
\end{array}\right]               
$
Solution:
Method 1: Using elementary row operations.
• We can write a general procedure:
1. We are given a 3×3 matrix.
• So the diagonal elements are: a₁₁, a₂₂ and a₃₃. These elements must be ‘1’ in the final answer.
2. First we change a₁₁ to 1.
• Then we change the remaining elements in that column, to zeros. We do that step by step.
3. Next, we change a₂₂ to 1.
• Then we change the remaining elements in that column, to zeros. We do that step by step.
4. Finally, we change a₃₃ to 1.
• Then we change the remaining elements in that column, to zeros. We do that step by step.

Method 2: Using elementary column operations.
• We can write a general procedure:
1. We are given a 3×3 matrix.
• So the diagonal elements are: a₁₁, a₂₂ and a₃₃. These elements must be ‘1’ in the final answer.
2. First we change a₁₁ to 1.
• Then we change the remaining elements in that row, to zeros. We do that step by step.
3. Next, we change a₂₂ to 1.
• Then we change the remaining elements in that row, to zeros. We do that step by step.
4. Finally, we change a₃₃ to 1.
• Then we change the remaining elements in that row, to zeros. We do that step by step.

Solved Example 19.18
By using elementary operations, find the inverse of the matrix A = $\left[\begin{array}{r}               
10    &{    -2    }    \\
-5    &{    1    }    \\
\end{array}\right]               
$
Solution:
Method 1: Using elementary row operations.

• We see that:
In (4) we did an elementary operation. As a result of that single operation, all elements of a row became zeroes.
• So for the given matrix A, the inverse does not exist.

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The link below gives a few more solved examples:

Exercise 19.4

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In the next section, we will see some miscellaneous examples.

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