If D = left| {begin{array}{*{20}{c}} {{a₁}}&{{b₁}}&{{c₁}} {{a₂}

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3 and 4 .Determinants and Matrices

normal

If $D = \left| {\begin{array}{{20}{c}} {{a_1}}&{{b_1}}&{{c_1}} \\ {{a_2}}&{{b_2}}&{{c_2}} \\ {{a_3}}&{{b_3}}&{{c_3}} \end{array}} \right|$ and $D' = \left| {\begin{array}{{20}{c}} {{a_1} + p{b_1}}&{{b_1} + q{c_1}}&{{c_1} + r{a_1}} \\ {{a_2} + p{b_2}}&{{b_2} + q{c_2}}&{{c_2} + r{a_2}} \\ {{a_3} + p{b_3}}&{{b_3} + q{c_3}}&{{c_3} + r{a_3}} \end{array}} \right|$ then

$D' = D$

$D' = (1 + pqr)D$

$D' = (1 -pqr)D$

$D' = pqrD$

Solution

$D = \left| {\begin{array}{*{20}{c}}
{{a_1} + p{b_1}}&{{b_1} + q{c_1}}&{r{a_1}} \\
{{a_2} + p{b_2}}&{{b_2} + q{c_2}}&{r{a_2}} \\
{{a_3} + p{b_3}}&{{b_3} + q{c_3}}&{r{a_3}}
\end{array}} \right| = D'$

$\Rightarrow$ $D' = (1 + pqr) D$

Standard 12

Mathematics

Express the following matrices as the sum of a symmetric and a skew symmetric matrix : $\left[\begin{array}{rrr}6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3\end{array}\right]$

medium

Out of the following a skew- symmetric matrix is

easy

Let $A=\left[\begin{array}{cc}\frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\ \frac{-3}{\sqrt{10}} & \frac{1}{\sqrt{10}}\end{array}\right]$ and $B =\left[\begin{array}{rr}1 & – i \\ 0 & 1\end{array}\right]$, where $i =\sqrt{-1}$. If $M = A ^{ T } BA$, then the inverse of the matrix $AM ^{2023} A ^{ T }$ is $………$

hard

(JEE MAIN-2023)

Express the following matrices as the sum of a symmetric and a skew symmetric matrix : $\left[\begin{array}{ccc}3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2\end{array}\right]$

medium

Let $A=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}9^{2} & -10^{2} & 11^{2} \\ 12^{2} & 13^{2} & -14^{2} \\ -15^{2} & 16^{2} & 17^{2}\end{array}\right]$, then the value of $A ^{\prime} BA$ is.

medium

(JEE MAIN-2022)

Solution

Similar Questions

Express the following matrices as the sum of a symmetric and a skew symmetric matrix : $\left[\begin{array}{rrr}6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3\end{array}\right]$

Out of the following a skew- symmetric matrix is

Express the following matrices as the sum of a symmetric and a skew symmetric matrix : $\left[\begin{array}{ccc}3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2\end{array}\right]$

Let $A=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}9^{2} & -10^{2} & 11^{2} \\ 12^{2} & 13^{2} & -14^{2} \\ -15^{2} & 16^{2} & 17^{2}\end{array}\right]$, then the value of $A ^{\prime} BA$ is.

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