Question

Let P₁ = I = [(1, 0, 0),(0, 1, 0),(0, 0, 1)], P₂ = [(1, 0, 0), (0, 0, 1), (0, 1, 0)], P₃ = [(0, 1, 0), (1, 0, 0), (0, 0, 1)], P₄ = [(0, 1, 0), (0, 0, 1), (1, 0, 0)], P₅ = [(0, 0, 1), (1, 0, 0), (0, 1, 0), P₆ = [(0, 0, 1), (0, 1, 0), (1, 0 , 0)] and x = ∑ k ∈ [k = 1, 6] P_k [(2, 1, 3), (1, 0, 2), (3, 2, 1)] P^T_k 

where P^T_k denotes the transpose of the matrix P_k. Then which of the following options is/are correct? 

(A) The sum of diagonal entries of X is 18

(C) X is a symmetric matrix 

(D) X – 30 I is an invertible matrix 

Answer 1

Correct option : (A, B, C)

Explanation: 

Comments

How option b is correct

Answer 2

Answer is (A,B,C)

(A) The sum of diagonal entries of X is 18

Sum of diagonal elements is known as trace. 

Now,

So, sum of diagonal elements of X is equal to 18.

(B) If \(X\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\) \(=\alpha \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\) then α = 30

It implies that α = 30.

Hence,  If \(X\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\) \(=\alpha \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}\) then α = 30

(C) X is a symmetric matrix 

So, X is a symmetric matrix.

(D) X – 30 I is an invertible matrix 

We know that,

It implies that X - 30 I is non-invertible matrix.

Hence, the correct answer is option (A), (B) and (C).