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Matrices and Determinants question

2019 · Shift 2 · Q26
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  5. /2019 · Shift 2 · Q26

Matrices and Determinants question

2019 · Shift 2 · Q26

JEE AdvancedMathematicsMatrices and DeterminantsMultiple correct+4 / −1
P1=I=[100010001], P2=[100001010], P3=[010100001], P4=[010001100], P5=[001100010], P6=[001010100]{P_1} = I = \left[ {\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} } \right],\,{P_2} = \left[ {\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ \end{matrix} } \right],\,{P_3} = \left[ {\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{matrix} } \right],\,{P_4} = \left[ {\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \\ \end{matrix} } \right],\,{P_5} = \left[ {\begin{matrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{matrix} } \right],\,{P_6} = \left[ {\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ \end{matrix} } \right]P1​=I=​100​010​001​​,P2​=​100​001​010​​,P3​=​010​100​001​​,P4​=​001​100​010​​,P5​=​010​001​100​​,P6​=​001​010​100​​ and X=∑k=16Pk[213102321]PkTX = \sum\limits_{k = 1}^6 {{P_k}} \left[ {\begin{matrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1 \\ \end{matrix} } \right]P_k^TX=k=1∑6​Pk​​213​102​321​​PkT​ where PkTP_k^TPkT​ denotes the transpose of the matrix Pk. Then which of the following option is/are correct?
  1. A
    X is a symmetric matrix
  2. B
    The sum of diagonal entries of X is 18
  3. C
    X −-− 30I is an invertible matrix
  4. D
    If X[111]=α[111]X\left[ {\begin{matrix} 1 \\ 1 \\ 1 \\ \end{matrix} } \right] = \alpha \left[ {\begin{matrix} 1 \\ 1 \\ 1 \\ \end{matrix} } \right]X​111​​=α​111​​, then α=30\alpha = 30α=30
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