Question

Which of the following statements is/are correct?
I. Adjoint of a unit matrix is a unit matrix.
II. \( A (\text{adj } A) = (\text{adj } A) A = |A|I \)
III. Adjoint of a symmetric matrix is symmetric.
IV. Adjoint of a diagonal matrix is a diagonal matrix.

• A. I
• B. II
• C. I or II
• D. All statements are correct

Hint:

The adjoint of a unit matrix is a unit matrix, and the adjoint of a symmetric or diagonal matrix retains the symmetry or diagonal structure.

Answer 1

The Correct Option is D

Step 1: Understanding the adjoint of a unit matrix.
The adjoint of a matrix is the transpose of its cofactor matrix. The cofactor matrix of a unit matrix is also a unit matrix because each minor of the unit matrix is either 1 or 0, and its cofactor matrix has the same structure as the original matrix. Thus, the adjoint of a unit matrix is a unit matrix. Hence, statement I is correct.

Step 2: Formula for \( A (\text{adj } A) = (\text{adj } A) A = |A|I \).
For any square matrix \( A \), it is a well-known property that: \[ A (\text{adj } A) = (\text{adj } A) A = |A|I \] where \( \text{adj } A \) is the adjoint of \( A \), and \( |A| \) is the determinant of \( A \). This is a standard result in linear algebra. Thus, statement II is correct.

Step 3: Adjoint of a symmetric matrix.
The adjoint of a symmetric matrix is also symmetric because the cofactor matrix, which is used to form the adjoint, will also maintain symmetry if the original matrix is symmetric. Thus, statement III is correct.

Step 4: Adjoint of a diagonal matrix.
The adjoint of a diagonal matrix is also diagonal because each cofactor corresponding to a diagonal element is non-zero, while all off-diagonal elements will have a cofactor of zero. Thus, statement IV is correct.

Step 5: Conclusion.
Since all four statements are correct, the correct answer is option (4), "All statements are correct."