Question

If \( A = \begin{bmatrix} 3 & 3 & 1 \\ 3 & 4 & 1 \\ 4 & 3 & 1 \end{bmatrix} \), then verify that \( A \, (\text{adj } A) = |A| I \) and find \( A^{-1} \).

Hint:

The relationship A(adj A) = \(|\)A\(|\)I is a fundamental theorem in matrix algebra. It provides a direct method to calculate the inverse of a matrix. Always start by calculating the determinant; if it's zero, the matrix is singular, and the inverse does not exist.

Answer 1

Step 1: Understanding the Concept:
This question involves two key properties of matrices. First, the relationship between a square matrix, its adjugate (adj A), its determinant (|A|), and the identity matrix (I), which is given by A(adj A) = (adj A)A = |A| I. Second, using this property to find the inverse of the matrix A, given by the formula A^-1 = (1/|A|) (adj A).
Step 2: Key Formula or Approach:
1. Calculate the determinant of A, |A|.
2. Find the matrix of cofactors of A.
3. Find the adjugate of A, adj(A), which is the transpose of the cofactor matrix.
4. Calculate the product A(adj A).
5. Calculate the product |A|I and verify it is equal to A(adj A).
6. Calculate the inverse A^-1 using the formula A^-1 = (1/|A|) adj(A).
Step 3: Detailed Explanation or Calculation:
1. Calculate the Determinant |A|:
|A| = 3(4·1 - 1·3) - 3(3·1 - 1·4) + 1(3·3 - 4·4)
|A| = 3(1) - 3(-1) + 1(-7) = 3 + 3 - 7 = -1
Since |A| ≠ 0, the inverse of A exists.
2. Find the Adjugate, adj(A):
Cofactor matrix C = [[1, 1, -7], [0, -1, 3], [-1, 0, 3]]
Adjugate is transpose of C:
adj(A) = [[1, 0, -1], [1, -1, 0], [-7, 3, 3]]
3. Verify A(adj A) = |A|I:
A(adj A) = [[3,3,1],[3,4,1],[4,3,1]] · [[1,0,-1],[1,-1,0],[-7,3,3]]
= [[-1,0,0],[0,-1,0],[0,0,-1]] = |A| I
4. Find A^-1:
A^-1 = (1/|A|) adj(A) = (-1)·[[1,0,-1],[1,-1,0],[-7,3,3]]
A^-1 = [[-1,0,1],[-1,1,0],[7,-3,-3]]
Step 4: Final Answer:
The property A(adj A) = |A|I is verified.
The inverse of the matrix A is A^-1 = [[-1,0,1],[-1,1,0],[7,-3,-3]].