To solve the problem of finding the adjugate of the matrix [tex]\( A = \begin{pmatrix} 2 & 3 \\ 4 & -5 \end{pmatrix} \)[/tex], we can proceed with the following steps:
### Step 1: Calculate the Determinant of the Matrix [tex]\( A \)[/tex]
The determinant of a 2x2 matrix [tex]\( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \)[/tex] is given by the formula:
[tex]\[
\text{Det}(A) = ad - bc
\][/tex]
For the given matrix [tex]\( A \)[/tex]:
[tex]\( a = 2 \)[/tex], [tex]\( b = 3 \)[/tex], [tex]\( c = 4 \)[/tex], and [tex]\( d = -5 \)[/tex]. Therefore,
[tex]\[
\text{Det}(A) = (2 \cdot (-5)) - (3 \cdot 4) = -10 - 12 = -22
\][/tex]
So, the determinant of the matrix [tex]\( A \)[/tex] is [tex]\(-22\)[/tex].
### Step 2: Confirm the Matrix is Invertible
A matrix is invertible if its determinant is non-zero. Since the determinant of [tex]\( A \)[/tex] is [tex]\(-22\)[/tex], which is non-zero, the matrix [tex]\( A \)[/tex] is indeed invertible.
### Step 3: Compute the Adjugate of the Matrix [tex]\( A \)[/tex]
For a 2x2 matrix [tex]\( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \)[/tex], the adjugate is computed by:
[tex]\[
\text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\][/tex]
Applying this to our matrix [tex]\( A \)[/tex]:
[tex]\[
\text{adj}(A) = \begin{pmatrix} -5 & -3 \\ -4 & 2 \end{pmatrix}
\][/tex]
### Step 4: Validate the Result
We can verify our result for the adjugate is correct based on the solution provided:
- The computed determinant is [tex]\(-22\)[/tex].
- The computed adjugate matrix entries are [tex]\(-5\)[/tex], [tex]\(-3\)[/tex], [tex]\(-4\)[/tex], and [tex]\(2\)[/tex].
Thus, the final step-by-step solution to find the adjugate of [tex]\( A = \begin{pmatrix} 2 & 3 \\ 4 & -5 \end{pmatrix} \)[/tex] confirms:
- The determinant of [tex]\( A \)[/tex] is [tex]\(-22\)[/tex].
- The adjugate of [tex]\( A \)[/tex] is [tex]\( \begin{pmatrix} -5 & -3 \\ -4 & 2 \end{pmatrix} \)[/tex].
This matches the given solution.
