cmi-entrance 2022 QA3

cmi-entrance · India · ugmath_23may 4 marks Matrices Matrix Entry and Coefficient Identities
$M$ is a $3 \times 3$ matrix with integer entries. For $M$ we have (Sum of column 2) $= 4 \times$ (sum of column 1). (Sum of column 3) $= 4 \times$ (sum of column 2). (Sum of row $2) = 6 +$ (sum of row $1$). (Sum of row $3) = 6 +$ (sum of row 2).
Statements
(9) The sum of all the entries in $M$ must be divisible by 21. (10) None of the row sums is divisible by 7. (11) One of the column sums must be divisible by 7. (12) None of the column sums is divisible by 6.
$M$ is a $3 \times 3$ matrix with integer entries. For $M$ we have\\
(Sum of column 2) $= 4 \times$ (sum of column 1). (Sum of column 3) $= 4 \times$ (sum of column 2).\\
(Sum of row $2) = 6 +$ (sum of row $1$).\\
(Sum of row $3) = 6 +$ (sum of row 2).

\textbf{Statements}

(9) The sum of all the entries in $M$ must be divisible by 21.\\
(10) None of the row sums is divisible by 7.\\
(11) One of the column sums must be divisible by 7.\\
(12) None of the column sums is divisible by 6.