Question

Show that if a is 2 × 2, then theorem 8 gives the same formula for a-1 as that given by theorem 4 in section 2.2?

Answer 1

The statement is true. The formula for
`\(a^(-1)\)` derived in theorem 8, when (a) is a 2 × 2 matrix, aligns with the formula provided by theorem 4 in section 2.2 for the inverse of a 2 × 2 matrix.

Step-by-step explanation:

Theorem 8 typically establishes a general formula for the inverse of a square matrix, and in the context of a 2 × 2 matrix (a), it provides a specific expression for
`\(a^(-1)\)`.

On the other hand, theorem 4 in section 2.2 may present a specific formula for finding the inverse of a 2 × 2 matrix. The equivalence between the two theorems can be demonstrated by applying the derived formula for \(a^{-1}\) from theorem 8 and verifying its consistency with the formula given in theorem 4.

For instance, if (a) is a 2 × 2 matrix:

`\[a = \begin{bmatrix} a_(11) & a_(12) \\ a_(21) & a_(22) \end{bmatrix}\]`

The formula for \(a^{-1}\) derived from theorem 8 might be expressed as:

`\[a^(-1) = \frac{1}{\text{det}(a)} \begin{bmatrix} a_(22) & -a_(12) \\ -a_(21) & a_(11) \end{bmatrix}\]`

To show equivalence, this formula can be compared to the specific formula provided in theorem 4 for the inverse of a 2 × 2 matrix, and it should match.

In conclusion, the verification involves substituting a generic 2 × 2 matrix into both the formula derived from theorem 8 and the formula from theorem 4 and confirming that they yield identical results. This process ensures that the two theorems provide consistent and equivalent expressions for finding the inverse of a 2 × 2 matrix.