Total number of matrices $A = left[ {begin{array}{*{20}{c}} 0&{2x}&{2x} {2y}&y&

English

Gujarati

Hindi

3 and 4 .Determinants and Matrices

hard

The total number of matrices $A = \left[ {\begin{array}{*{20}{c}}
0&{2x}&{2x}\\
{2y}&y&{ - y}\\
1&{ - 1}&1
\end{array}} \right];\,\left( {x,y \in R,\,x \ne y} \right)$ for which ${A^T}A = 3{I_3}$

$6$

$2$

$3$

$4$

(JEE MAIN-2019)

Solution

${A^T}A = 3{I_3}$

$ \Rightarrow \left[ {\begin{array}{*{20}{c}}
0&{2x}&{2x}\\
{2y}&y&{ – y}\\
1&{ – 1}&1
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
0&{2y}&1\\
{2x}&y&{ – 1}\\
{2x}&{ – y}&1
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
3&0&0\\
0&3&0\\
0&0&3
\end{array}} \right]$

$ \Rightarrow \left[ {\begin{array}{*{20}{c}}
{8{x^2}}&0&0\\
0&{6{y^2}}&0\\
0&0&3
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
3&0&0\\
0&3&0\\
0&0&3
\end{array}} \right]$

$ \Rightarrow 8{x^2} = 3 \Rightarrow x = \pm \sqrt {\frac{3}{8}} $

$ \Rightarrow 6{y^2} = 3 \Rightarrow y = \pm \sqrt {\frac{1}{2}} $

$4$ matrices are possible

Standard 12

Mathematics

If $\mathrm{A}, \,\mathrm{B}$ are symmetric matrices of same order, then $\mathrm{A B}-\mathrm{B A}$ is a

easy

Matrix $\left[ {\begin{array}{*{20}{c}}0&{ – 4}&1\\4&0&{ – 5}\\{ – 1}&5&0\end{array}} \right]$is

easy

Let $A$ be a $2 \times 2$ real matrix with entries from $\{0,1\}$ and $|\mathrm{A}| \neq 0 .$ Consider the following two statements :

$(P)$ If $A \neq I_{2},$ then $|A|=-1$

$(\mathrm{Q})$ If $|\mathrm{A}|=1,$ then $\operatorname{tr}(\mathrm{A})=2$

where $I_{2}$ denotes $2 \times 2$ identity matrix and $\operatorname{tr}(A)$ denotes the sum of the diagonal entries of $A$ Then

hard

(JEE MAIN-2020)

Which of the following is incorrect

medium

Characteristic equation of $A = \left| {\begin{array}{*{20}{c}}
2&3&0\\
1&2&5\\
3&{ – 1}&2
\end{array}} \right|$ is

normal

The total number of matrices $A = \left[ {\begin{array}{*{20}{c}} 0&{2x}&{2x}\\ {2y}&y&{ - y}\\ 1&{ - 1}&1 \end{array}} \right];\,\left( {x,y \in R,\,x \ne y} \right)$ for which ${A^T}A = 3{I_3}$

Solution

Similar Questions

If $\mathrm{A}, \,\mathrm{B}$ are symmetric matrices of same order, then $\mathrm{A B}-\mathrm{B A}$ is a

Matrix $\left[ {\begin{array}{*{20}{c}}0&{ – 4}&1\\4&0&{ – 5}\\{ – 1}&5&0\end{array}} \right]$is

Which of the following is incorrect

Characteristic equation of $A = \left| {\begin{array}{*{20}{c}} 2&3&0\\ 1&2&5\\ 3&{ – 1}&2 \end{array}} \right|$ is

Start a Free Trial Now

The total number of matrices $A = \left[ {\begin{array}{*{20}{c}}
0&{2x}&{2x}\\
{2y}&y&{ - y}\\
1&{ - 1}&1
\end{array}} \right];\,\left( {x,y \in R,\,x \ne y} \right)$ for which ${A^T}A = 3{I_3}$

Characteristic equation of $A = \left| {\begin{array}{*{20}{c}}
2&3&0\\
1&2&5\\
3&{ – 1}&2
\end{array}} \right|$ is