Let M be any 3 × 3 matrix with entries from set {0,1,2} . maximum number of such matrices, for whic

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3 and 4 .Determinants and Matrices

medium

Let $M$ be any $3 \times 3$ matrix with entries from the set $\{0,1,2\}$. The maximum number of such matrices, for which the sum of diagonal elements of $M ^{ T } M$ is seven, is .............

$512$

$556$

$560$

$540$

(JEE MAIN-2021)

Solution

$\left[\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array}\right]\left[\begin{array}{lll}a & d & g \\ b & e & h \\ c & f & i\end{array}\right]$

$a^{2}+b^{2}+c^{2}+d^{2}+e^{2}+f^{2}+g^{2}+h^{2}+i^{2}=7$

Case$-I$ : Seven $(1's)$ and two $(0's)$

${ }^{9} C _{2}=36$

Case$-II$ : One $(2)$ and three $(1's)$ and five $(0's)$

$\frac{9 !}{5 ! 3 !}=504$

$\therefore$ Total $=540$

Standard 12

Mathematics

If $A = \left[ {\begin{array}{*{20}{c}}{\cos \theta }&{ – \sin \theta }\\{\sin \theta }&{\cos \theta }\end{array}} \right]$, then which of the following statements is not correct

easy

Let $A$ is a symmetric and $ \,B$ is a skew symmetric matrix, such that $A – B = \left[ {\begin{array}{*{20}{c}}
1&2 \\
3&4
\end{array}} \right]$, then $|A|$ is

normal

Let three matrices $A =$$\left[ {\begin{array}{{20}{c}}2&1\\4&1\end{array}} \right]$ ; $B =$$\left[ {\begin{array}{{20}{c}}3&4\\2&3\end{array}} \right]$ and $C =$$\left[ {\begin{array}{*{20}{c}}3&{ – 4}\\{ – 2}&3\end{array}} \right]$ then $T_r(A) + t_r$ $\left( {\frac{{ABC}}{2}} \right)$ $+$ $t_r$$\left( {\frac{{A{{(BC)}^2}}}{4}} \right)$ $+$ $t_r$ $\left( {\frac{{A{{(BC)}^3}}}{8}} \right)$ $+$ ……. $+ \infty =$

normal

If $A$ and $B$ are symmetric matrices of the same order, then show that $AB$ is symmetric if and only if $A$ and $B$ commute, that is $AB = BA$.

medium

The matrix $\left[ {\begin{array}{*{20}{c}}0&5&{ – 7}\\{ – 5}&0&{11}\\7&{ – 11}&0\end{array}} \right]$is known as