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GO Classes DPP | GATE CS, DA | Linear Algebra | Determinant

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Consider a $4 \times 4$ matrix $M$. Suppose that the following four entries of the matrix are known to be zero.

For which of the following cases must the determinant of the matrix be equal to $0$ ?

  1. $M[0,0]=0, ~M[1,1]=0, ~M[2,2]=0, ~M[3,3]=0$
     
  2. $M[0,0]=0, ~M[0,1]=0, ~M[0,2]=0, ~M[0,3]=0$
     
  3. $M[1,0]=0, ~M[0,1]=0, ~M[3,1]=0, ~M[2,3]=0$
     
  4. $M[0,2]=0, ~M[1,2]=0, ~M[2,2]=0, ~M[3,2]=0$

2 Answers

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Recall that the determinant of a matrix is zero if the matrix has an entire row or an entire column consisting of zeros.

We analyze each option:

  1. The entries $M[0,0], M[1,1], M[2,2], M[3,3]$ are zero.  
    These are diagonal entries only, and do not force an entire row or column to be zero.  
    Therefore, the determinant is not necessarily zero.
     
  2. The entries $M[0,0], M[0,1], M[0,2], M[0,3]$ are zero.  
    This means the entire first row is zero.  
    Hence, $\det(M) = 0$.
     
  3. The entries $M[1,0], M[0,1], M[3,1], M[2,3]$ are zero.  
    These entries are scattered and do not form a complete row or column.  
    Therefore, the determinant is \emph{not necessarily} zero.
     
  4. The entries $M[0,2], M[1,2], M[2,2], M[3,2]$ are zero.  
    This means the entire third column is zero.  
    Hence, $\det(M) = 0$.
     

Answer: $\boxed{\text{B and D}}$

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