Matrix

We know the identity:
A ⋅ adj(A) = |A| ⋅ I
⇒ |A| ⋅ I = [ -2 0 0 ]
[ 0 -2 0 ]
[ 0 0 -2 ]
⇒ |A| = -2
Also, for a 3 × 3 matrix,
|adj(A)| = |A|^(n-1) = |A|² = (-2)² = 4

To determine the values of that make matrix singular:

Given Matrix:

1. Singularity Condition:
A matrix is singular when its determinant equals zero:

2. Compute the Determinant:

3. Expand and Simplify:

4. Solve the Equation:
Set the determinant to zero and solve for :

Final Solution:
The matrix is singular when:

Given the matrix equation:

1. Apply Fundamental Matrix Identity:
We know from matrix theory that:

2. Equate Both Expressions:
Comparing with the given equation:

3. Determine Order of Matrix A:
The problem implies A is 4×4 (as evident from the context). For an n×n matrix:

4. Calculate Adjugate Determinant:
For n = 4:

Final Result:

To find the inverse of matrix , we follow these steps:

1. Calculate the Determinant:

2. Find the Matrix of Cofactors:

3. Form the Adjugate Matrix:
Transpose the cofactor matrix:

4. Compute the Inverse:
Using the formula :

Key Verification:
We can verify by checking :