Answer
456.6k+ views
Hint: Expanding the determinant yields an equation in terms of $x$, which then can be solved to determine the value of $x$.
The given determinant is $\left| {\begin{array}{*{20}{c}}
x&2&0 \\
2&{ - x}&0 \\
0&3&x
\end{array}} \right| = 0$
We need to expand the determinant to get an equation.
While expanding a determinant, the signs are as follows,
$\left| {\begin{array}{*{20}{c}}
+ & - & + \\
- & + & - \\
+ & - & +
\end{array}} \right|$
Choose a row or a column and start taking out one term after another and multiplying it with the
minors along with the corresponding sign.
When a term is taken out, the determinant leaving out the row and column of the term taken out
is called the minor.
Let us expand $\left| {\begin{array}{*{20}{c}}
x&2&0 \\
2&{ - x}&0 \\
0&3&x
\end{array}} \right| = 0$
$ + x\left| {\begin{array}{*{20}{c}}
{ - x}&0 \\
3&x
\end{array}} \right| - 2\left| {\begin{array}{*{20}{c}}
2&0 \\
0&x
\end{array}} \right| + 0\left| {\begin{array}{*{20}{c}}
2&{ - x} \\
0&3
\end{array}} \right| = 0$
Now we need to expand the $2 \times 2$determinant,
$x( - x(x) - 0(3)) - 2(2(x) - 0(0)) + 0 = 0$
$\begin{gathered}
x( - {x^2} - 0) - 2(2x - 0) + 0 = 0 \\
- {x^3} - 4x = 0 \\
{x^3} + 4x = 0 \\
x({x^2} + 4) = 0 \\
x = 0,{x^2} = - 4 \\
\end{gathered} $
${x^2} = - 4$ is not possible because it yields imaginary roots.
$x = 0$ is the correct answer.
Note: Expanding the determinant yields an equation in terms of $x$, which then can be solved to determine the value of $x$. The signs, while expanding the determinant can be tricky.
The given determinant is $\left| {\begin{array}{*{20}{c}}
x&2&0 \\
2&{ - x}&0 \\
0&3&x
\end{array}} \right| = 0$
We need to expand the determinant to get an equation.
While expanding a determinant, the signs are as follows,
$\left| {\begin{array}{*{20}{c}}
+ & - & + \\
- & + & - \\
+ & - & +
\end{array}} \right|$
Choose a row or a column and start taking out one term after another and multiplying it with the
minors along with the corresponding sign.
When a term is taken out, the determinant leaving out the row and column of the term taken out
is called the minor.
Let us expand $\left| {\begin{array}{*{20}{c}}
x&2&0 \\
2&{ - x}&0 \\
0&3&x
\end{array}} \right| = 0$
$ + x\left| {\begin{array}{*{20}{c}}
{ - x}&0 \\
3&x
\end{array}} \right| - 2\left| {\begin{array}{*{20}{c}}
2&0 \\
0&x
\end{array}} \right| + 0\left| {\begin{array}{*{20}{c}}
2&{ - x} \\
0&3
\end{array}} \right| = 0$
Now we need to expand the $2 \times 2$determinant,
$x( - x(x) - 0(3)) - 2(2(x) - 0(0)) + 0 = 0$
$\begin{gathered}
x( - {x^2} - 0) - 2(2x - 0) + 0 = 0 \\
- {x^3} - 4x = 0 \\
{x^3} + 4x = 0 \\
x({x^2} + 4) = 0 \\
x = 0,{x^2} = - 4 \\
\end{gathered} $
${x^2} = - 4$ is not possible because it yields imaginary roots.
$x = 0$ is the correct answer.
Note: Expanding the determinant yields an equation in terms of $x$, which then can be solved to determine the value of $x$. The signs, while expanding the determinant can be tricky.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)