Question

Consider the matrices $A = \begin{bmatrix} 1 & 1 & 1 \\ a & b & c \\ bc & ca & ab \end{bmatrix}$ and $B = \begin{bmatrix} a(b^2-c^2) & a(c-b) & (c-b) \\ b(c^2-a^2) & b(a-c) & (a-c) \\ c(a^2-b^2) & c(b-a) & (b-a) \end{bmatrix}$. Which of the following statement(s) is(are) correct?

• A. The product of $A$ and $B$ is a diagonal matrix
• B. The product of $A$ and $B$ is a symmetric matrix
• C. Trace of $AB$ is equal to $3|A|$

Answer 1

Answer: (A), (B), (C)

A, B, C

Answer 2

To determine which statements are correct, we need to analyze the product of matrices $A$ and $B$.

1. Calculate the determinant of matrix A:

   $|A| = \begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ bc & ca & ab \end{vmatrix} = (a-b)(b-c)(c-a)$

2. Calculate the product matrix $C = AB$.

   $C_{11} = 1 \cdot a(b^2-c^2) + 1 \cdot b(c^2-a^2) + 1 \cdot c(a^2-b^2) = (a-b)(b-c)(c-a) = |A|$.
   $C_{12} = 1 \cdot a(c-b) + 1 \cdot b(a-c) + 1 \cdot c(b-a) = 0$.
   $C_{13} = 1 \cdot (c-b) + 1 \cdot (a-c) + 1 \cdot (b-a) = 0$.
   $C_{21} = a \cdot a(b^2-c^2) + b \cdot b(c^2-a^2) + c \cdot c(a^2-b^2) = 0$.
   $C_{22} = a \cdot a(c-b) + b \cdot b(a-c) + c \cdot c(b-a) = (a-b)(b-c)(c-a) = |A|$.
   $C_{23} = a \cdot (c-b) + b \cdot (a-c) + c \cdot (b-a) = 0$.
   $C_{31} = bc \cdot a(b^2-c^2) + ca \cdot b(c^2-a^2) + ab \cdot c(a^2-b^2) = 0$.
   $C_{32} = bc \cdot a(c-b) + ca \cdot b(a-c) + ab \cdot c(b-a) = 0$.
   $C_{33} = bc \cdot (c-b) + ca \cdot (a-c) + ab \cdot (b-a) = (a-b)(b-c)(c-a) = |A|$.

3. The product matrix is $AB = \begin{bmatrix} |A| & 0 & 0 \\ 0 & |A| & 0 \\ 0 & 0 & |A| \end{bmatrix} = |A| I_3$.

4. Evaluate the statements:

   • A. $AB$ is a diagonal matrix. (Correct, as it is of the form $kI$)
   • B. $AB$ is a symmetric matrix. (Correct, as all diagonal matrices are symmetric)
   • C. Trace($AB$) $= |A| + |A| + |A| = 3|A|$. (Correct)

All statements A, B, and C are correct.