Question

Which of the following statements is false?

• A. If \( | A | = 0 \), then \( | \text{adj } A | = 0 \)
• B. Adjoint of a diagonal matrix of order \( 3 \times 3 \) is a diagonal matrix
• C. The product of two upper triangular matrices is an upper triangular matrix
• D. \( \text{adj} (AB) = \text{adj} (A) \text{adj} (B) \)

Hint:

Remember key properties of adjoints: - \( |\text{adj } A| = |A|^{n-1} \) - \( \text{adj} (AB) = \text{adj} (B) \text{adj} (A) \) (Note the reversal of order) - Adjoint of a diagonal matrix is diagonal. - Adjoint of a symmetric matrix is symmetric.

Answer 1

The Correct Option is D

Step 1: Analyze statement (A). We know the property relating the determinant of the adjoint of a matrix A (of order n) to the determinant of A: \( |\text{adj } A| = |A|^{n-1} \). For a \( 3 \times 3 \) matrix, \( n=3 \), so \( |\text{adj } A| = |A|^{3-1} = |A|^2 \). If \( |A| = 0 \), then \( |\text{adj } A| = (0)^2 = 0 \). Therefore, statement (A) is TRUE. Step 2: Analyze statement (B). Let \( A = \begin{bmatrix} a & 0 & 0
0 & b & 0
0 & 0 & c \end{bmatrix} \) be a diagonal matrix of order 3. The cofactor matrix C is \( C = \begin{bmatrix} bc & 0 & 0
0 & ac & 0
0 & 0 & ab \end{bmatrix} \). The adjoint of A is the transpose of the cofactor matrix: \( \text{adj } A = C^T = \begin{bmatrix} bc & 0 & 0
0 & ac & 0
0 & 0 & ab \end{bmatrix} \). This is also a diagonal matrix. Therefore, statement (B) is TRUE. Step 3: Analyze statement (C). Let A and B be two \( n \times n \) upper triangular matrices. This means \( A_{ij} = 0 \) for \( i > j \) and \( B_{ij} = 0 \) for \( i > j \). Consider the element \( (AB)_{ij} \) where \( i > j \). \( (AB)_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj} \). The term \( A_{ik} \) is non-zero only if \( i \le k \). The term \( B_{kj} \) is non-zero only if \( k \le j \). For a term \( A_{ik} B_{kj} \) to be non-zero, we need \( i \le k \) and \( k \le j \), which implies \( i \le k \le j \). However, we are considering the case where \( i > j \). It is impossible to satisfy \( i \le k \le j \) when \( i > j \). Therefore, every term \( A_{ik} B_{kj} \) in the sum must be zero, meaning \( (AB)_{ij} = 0 \) for \( i > j \). Thus, the product AB is also an upper triangular matrix. Therefore, statement (C) is TRUE. Step 4: Analyze statement (D). Recall the property for the adjoint of a product of matrices: \[ \text{adj} (AB) = \text{adj} (B) \text{adj} (A) \] The statement given is \( \text{adj} (AB) = \text{adj} (A) \text{adj} (B) \). Since matrix multiplication (and thus multiplication of adjoints) is generally not commutative, the given statement is incorrect. Therefore, statement (D) is FALSE. Step 5: Identify the false statement. Based on the analysis, statement (D) is the false statement.