Matrices

For any square matrix of order , the determinant of its adjugate (or adjoint) is given by: This property comes from the fact that the adjugate matrix is formed by the cofactors, and its determinant is the original determinant raised to the power . Given: So, Therefore, the value of is 36.

Compute : So,

Let . Then, So, Solve: Next: So,

Step 1: Write the system in matrix form:
where
Step 2: Find the determinant of :
Step 3: Find the cofactor matrix and its transpose:
So, Step 4: Multiply :
Compute each entry:

Final Answer: