Find x, if [x-5-1]left[begin{array}{lll}1 & 0 & 2 0 & 2 & 1 2 & 0 & 3end{arr

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3 and 4 .Determinants and Matrices

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Find $x,$ if $[x-5-1]\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3\end{array}\right]\left[\begin{array}{l}x \\ 4 \\ 1\end{array}\right]=O$

$\pm 4 \sqrt{3}$

Solution

We have:

$\left[\begin{array}{ccc}x & -5 & -1\end{array}\right]\left[\begin{array}{ccc}1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3\end{array}\right]\left[\begin{array}{l}x \\ 4 \\ 1\end{array}\right]=0$

$\Rightarrow\left[\begin{array}{lll}x+0-2 & 0-10+0 & 2 x-5-3\end{array}\right]\left[\begin{array}{l}x \\ 4 \\ 1\end{array}\right]=0$

$\Rightarrow\left[\begin{array}{lll}x-2 & -10 & 2 x-8\end{array}\right]\left[\begin{array}{l}x \\ 4 \\ 1\end{array}\right]=0$

$\Rightarrow[x(x-2)-40+2 x-8]=0$

$\Rightarrow\left[x^{2}-2 x-40+2 x-8\right]=[0]$

$\Rightarrow\left[x^{2}-48\right]=[0]$

$\therefore x^{2}-48=0$

$\Rightarrow x^{2}=48$

$\Rightarrow x=\pm 4 \sqrt{3}$

Standard 12

Mathematics

Consider the following information regarding the number of men and women workers in three factories $I,\,II$ and $III$

Men workers
Women workers

$I$ $30$ $25$

$II$ $25$ $31$

$III$ $27$ $26$

Represent the above information in the form of a $3 \times 2$ matrix. What does the entry in the third row and second column represent?

easy

If the matrix $\left[ {\begin{array}{*{20}{c}}0&1&{ – 2}\\{ – 1}&0&3\\\lambda &{ – 3}&0\end{array}} \right]$ is singular, then $\lambda $=

easy

If $B = \left[ {\begin{array}{*{20}{c}}
5&{2\alpha }&1\\
0&2&1\\
\alpha &3&{ – 1}
\end{array}} \right]$ is the inverse of a $3 \times 3$ matrix $A$, then the sum of all values of $\alpha $ for which $det\, (A) + 1 = 0$, is

hard

(JEE MAIN-2019)

Let $A = \left( {\begin{array}{*{20}{c}}0&0&{ – 1}\\0&{ – 1}&0\\{ – 1}&0&0\end{array}} \right)$, the only correct statement about the matrix $A$ is

easy

(AIEEE-2004)

If $P = \left( {\begin{array}{{20}{c}}i&0&{ – i}\\0&{ – i}&i\\{ – i}&i&0\end{array}} \right)$ and $Q = \left( {\begin{array}{{20}{c}}{ – i}&i\\0&0\\i&{ – i}\end{array}} \right)$,then $PQ$ is equal to

easy

	Men workers	Women workers
$I$	$30$	$25$
$II$	$25$	$31$
$III$	$27$	$26$

Find $x,$ if $[x-5-1]\left[\begin{array}{lll}1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3\end{array}\right]\left[\begin{array}{l}x \\ 4 \\ 1\end{array}\right]=O$

Solution

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5&{2\alpha }&1\\
0&2&1\\
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\end{array}} \right]$ is the inverse of a $3 \times 3$ matrix $A$, then the sum of all values of $\alpha $ for which $det\, (A) + 1 = 0$, is

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