Solve the following$\left| {\begin{array}{*{20}{c}}
x&2&0 \\
2&{ - x}&0 \\
0&3&x
\end{array}} \right| = 0$
Last updated date: 24th Mar 2023
•
Total views: 311.4k
•
Views today: 4.88k
Answer
311.4k+ 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
If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

KCN reacts readily to give a cyanide with A Ethyl alcohol class 12 chemistry JEE_Main

Trending doubts
What was the capital of Kanishka A Mathura B Purushapura class 7 social studies CBSE

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Tropic of Cancer passes through how many states? Name them.

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

Name the Largest and the Smallest Cell in the Human Body ?
