Statement I: For square matrices A and B of same order (with real entries) (O is null matrix and is Identity matrix of same order as A and B). If AB = O then Statement II: If denote the direction cosines of 3 mutual perpendicular vectors then is a

Question

Which of the following must be the true value of the statements below in that order.

Statement I: For square matrices A and B of same order (with real entries) (O is null matrix and $I$ is Identity matrix of same order as A and B). If AB = O then ${(\det A)}^{2} + {(\det B)}^{2} = 0$

Statement II: If $l_{i}, m_{i}, n_{i}$ $(i = 1, 2, 3)$ denote the direction cosines of 3 mutual perpendicular vectors then $A = (\begin{matrix} l_{1} & m_{1} & n_{1} \\ l_{2} & m_{2} & n_{2} \\ l_{3} & m_{3} & n_{3} \end{matrix})$ is an orthogonal matrix.

Statement III: For two distinct lines with shortest distance $d \neq 0$ we can have a point A on line one and point B on line two such that AB = d then A and B be unique points on them in all possible cases.

Smart Achievers · Accepted Answer

(3)

$A B = 0,$ $| B |$ $\neq 0,$ $A B = 0 \Rightarrow A = 0$

Solution

1 FFT

2 FFF

3 FTF

4 FTT

Ans.
(3)

$A B = 0,$ $| B |$ $\neq 0,$ $A B = 0 \Rightarrow A = 0$

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