If A = left[ begin{array}{l}123end{array} right], then AA' =

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3 and 4 .Determinants and Matrices

normal

If $A = \left[ \begin{array}{l}1\\2\\3\end{array} \right],$then $AA' = $

$[14]$

$\left[ \begin{array}{l}1\\4\\3\end{array} \right]$

$\left[ {\begin{array}{*{20}{c}}1&2&3\\2&4&6\\3&6&9\end{array}} \right]$

None of these

Solution

(c)$A' = [1\,\,2\,\,3]$ therefore $AA' = \left[ \begin{array}{l}1\\2\\3\end{array} \right]\,\,[1\,\,2\,\,3] = \left[ {\begin{array}{*{20}{c}}1&2&3\\2&4&6\\3&6&9\end{array}} \right]$.

Standard 12

Mathematics

Let $A$ and $B$ be any two $3 \times 3$ symmetric and skew symmetric matrices respectively. Then which of the following is $NOT$ true?

medium

(JEE MAIN-2022)

An orthogonal matrix is

easy

If $A=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right],$ then verify that $A^{\prime} A=I$

medium

If $\mathrm{A}=\left[\begin{array}{rrr}-1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{rrr}-4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1\end{array}\right],$ then verify that $(A+B)^{\prime}=A^{\prime}+B^{\prime}$.

easy

How many $3 \times 3$ matrices $\mathrm{M}$ with entries from $\{0,1,2\}$ are there, for which the sum of the diagonal entries of $M^T M$ is $5$ ?