UPSC NDA 2 2022 Mathematics Solved Paper

Concept:For matrix multiplication, an $m \times n$ matrix multiplied by an $p \times q$ matrix is defined only if $n = p$.The resulting matrix has order $m \times q$ and contains $m \times q$ entries.Explanation:Given: $X$ is $3 \times 3$, $Y$ is $2 \times 3$, $Z$ is $3 \times 2$.Statement 1: $(ZY)X$.Order of $Z$ is $3 \times 2$, order of $Y$ is $2 \times 3$.Since the inner dimensions match ($2 = 2$), $ZY$ is defined and has order $3 \times 3$.Now $(ZY)$ is $3 \times 3$ and $X$ is $3 \times 3$, so $(ZY)X$ is $3 \times 3$.It is a square matrix with $3 \times 3 = 9$ entries.Thus statement 1 is true.Statement 2: $Y(XZ)$.Order of $X$ is $3 \times 3$, order of $Z$ is $3 \times 2$, so $XZ$ has order $3 \times 2$.Now $Y$ is $2 \times 3$ and $XZ$ is $3 \times 2$, so $Y(XZ)$ is $2 \times 2$.It is a square matrix with $2 \times 2 = 4$ entries.Thus statement 2 is true.Statement 3: $X(YZ)$.Order of $Y$ is $2 \times 3$, order of $Z$ is $3 \times 2$, so $YZ$ has order $2 \times 2$.Now $X$ is $3 \times 3$ and $YZ$ is $2 \times 2$.Since $3 \neq 2$, the product $X(YZ)$ is not defined.Thus statement 3 is true.All three statements are correct.Answer:D. 1, 2 and 3