The given matrix $A$ is:

$A = \begin{pmatrix} -5 & 4 \\ -7 & 6 \end{pmatrix}.$

1. Determinant of $A$

The determinant of a $2 \times 2$ matrix is calculated as:

$\text{det}(A) = (a \cdot d) - (b \cdot c),$

where $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$.

For the given matrix:

$a = -5, \, b = 4, \, c = -7, \, d = 6.$

Substituting these values:

$\text{det}(A) = (-5 \cdot 6) - (4 \cdot -7) = -30 - (-28) = -30 + 28 = -2.$

So, the determinant of $A$ is:

$\text{det}(A) = -2.$

2. Adjoint of $A$

The adjoint (or adjugate) of a $2 \times 2$ matrix is found by swapping the diagonal elements and changing the signs of the off-diagonal elements. For $A = \begin{pmatrix} -5 & 4 \\ -7 & 6 \end{pmatrix}$:

$\text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}.$

Substituting $a = -5, \, b = 4, \, c = -7, \, d = 6$:

$\text{adj}(A) = \begin{pmatrix} 6 & -4 \\ 7 & -5 \end{pmatrix}.$

Thus, the adjoint of $A$ is:

$\text{adj}(A) = \begin{pmatrix} 6 & -4 \\ 7 & -5 \end{pmatrix}.$

Final Results:

1. Determinant: $-2$.
2. Adjoint: $\begin{pmatrix} 6 & -4 \\ 7 & -5 \end{pmatrix}$.

Would you like a step-by-step explanation of any part?
Here are some related questions for practice:

1. What is the inverse of $A$ based on its determinant and adjoint?
2. What are the eigenvalues of $A$?
3. How is the determinant affected if one row is scaled by a constant factor?
4. What happens to the adjoint if you transpose $A$?
5. Can you calculate the trace of the given matrix $A$?

Tip: For a $2 \times 2$ matrix, the determinant is a quick indicator of invertibility—if it's zero, the matrix is non-invertible.