Question

English

Gujarati

Hindi

3 and 4 .Determinants and Matrices

easy

If $A = \left[ {\begin{array}{{20}{c}}i&0\\0&{ - i}\end{array}} \right],B = \left[ {\begin{array}{{20}{c}}0&i\\i&0\end{array}} \right]$, where $i = \sqrt { - 1} $, then the correct relation is

$A + B = O$

${A^2} = {B^2}$

$A - B = O$

${A^2} + {B^2} = O$

Solution

(b) Relation ${A^2} = {B^2}$is true because ${A^2} = \left[ {\begin{array}{*{20}{c}}{ – 1}&0\\0&{ – 1}\end{array}} \right]$ and ${B^2} = \left[ {\begin{array}{*{20}{c}}{ – 1}&0\\0&{ – 1}\end{array}} \right]$have same matrices.

Std 12

Mathematics

If $A$ is square matrix such that $A^{2}=A$, then $(1+A)^{3}-7 A$ is equal to

medium

If $A,B$are square matrices of order $3, A$ is non- singular and $AB = O$, then $B$ is a

normal

For how many value $(s)$ of $x$ in the closed interval $[ – 4,\,\, – 1]$ is the matrix $\left[ {\begin{array}{*{20}{c}}3&{ – 1 + x}&2\\3&{ – 1}&{x + 2}\\{x + 3}&{ – 1}&2\end{array}} \right]$ singular

medium

Let $A = \,\left[ {\begin{array}{*{20}{c}}
1&0&0\\
1&1&0\\
1&1&1
\end{array}} \right]$ and $B = A^{20}$ . Then the sum of the elements of the first column of $B$ is?

hard

(JEE MAIN-2018)

The order of $[x\,y\,z]\,\,\left[ {\begin{array}{*{20}{c}}a&h&g\\h&b&f\\g&f&c\end{array}} \right]\,\left[ \begin{array}{l}x\\y\\z\end{array} \right]$ is