Answer
384.3k+ views
Hint: To solve this question we will use the properties of matrix. We will first simplify the given matrix by using the operations of matrix and determinants then we put the values obtained equal to zero, then we will find the value of $abc$ by simplifying the obtained equation.
Complete step by step solution:
We have been given that $a\ne 6,b,c$ satisfy $\left| \begin{matrix}
a & 2b & 2c \\
3 & b & c \\
4 & a & b \\
\end{matrix} \right|=0$.
We have to find the value of $abc$.
Now, let us first solve the given matrix. We know that we can solve the given matrix by multiplying the element by $2\times 2$ determinant. The determinant of a $3\times 3$ matrix is calculated for a matrix having 3 rows and 3 columns. Then we will get
$\Rightarrow a\left( b\times b-a\times c \right)-2b\left( 3\times b-4\times c \right)+2c\left( 3\times a-4\times b \right)$
Now, simplifying the above obtained equation we will get
$\begin{align}
& \Rightarrow a\left( {{b}^{2}}-ac \right)-2b\left( 3b-4c \right)+2c\left( 3a-4b \right) \\
& \Rightarrow a{{b}^{2}}-{{a}^{2}}c-6{{b}^{2}}+8bc+6ac-8bc \\
& \Rightarrow a{{b}^{2}}-{{a}^{2}}c-6{{b}^{2}}+6ac \\
\end{align}$
We have given that $a\ne 6,b,c$ satisfy matrix value equal to zero.
Then we will get
$\begin{align}
& \Rightarrow a{{b}^{2}}-{{a}^{2}}c-6{{b}^{2}}+6ac=0 \\
& \Rightarrow a{{b}^{2}}-6{{b}^{2}}-{{a}^{2}}c+6ac=0 \\
& \Rightarrow {{b}^{2}}\left( a-6 \right)-ac\left( a-6 \right)=0 \\
& \Rightarrow \left( a-6 \right)\left( {{b}^{2}}-ac \right)=0 \\
\end{align}$
If $a\ne 6$ then $\left( {{b}^{2}}-ac \right)=0$ then simplifying the obtained equation we will get
$\begin{align}
& \Rightarrow {{b}^{2}}-ac=0 \\
& \Rightarrow {{b}^{2}}=ac \\
& \Rightarrow abc={{b}^{3}} \\
\end{align}$
Hence we get the value of $abc$ as ${{b}^{3}}$.
Option C is the correct answer.
Note:
Students must remember the condition given in the question that $a\ne 6$. If we consider the factor $a-6=0$ then we will get the value $a=6$. So we need to ignore the factor. In matrices, determinants are the special numbers calculated from the square matrix. Square matrix should have an equal number of rows and columns.
Complete step by step solution:
We have been given that $a\ne 6,b,c$ satisfy $\left| \begin{matrix}
a & 2b & 2c \\
3 & b & c \\
4 & a & b \\
\end{matrix} \right|=0$.
We have to find the value of $abc$.
Now, let us first solve the given matrix. We know that we can solve the given matrix by multiplying the element by $2\times 2$ determinant. The determinant of a $3\times 3$ matrix is calculated for a matrix having 3 rows and 3 columns. Then we will get
$\Rightarrow a\left( b\times b-a\times c \right)-2b\left( 3\times b-4\times c \right)+2c\left( 3\times a-4\times b \right)$
Now, simplifying the above obtained equation we will get
$\begin{align}
& \Rightarrow a\left( {{b}^{2}}-ac \right)-2b\left( 3b-4c \right)+2c\left( 3a-4b \right) \\
& \Rightarrow a{{b}^{2}}-{{a}^{2}}c-6{{b}^{2}}+8bc+6ac-8bc \\
& \Rightarrow a{{b}^{2}}-{{a}^{2}}c-6{{b}^{2}}+6ac \\
\end{align}$
We have given that $a\ne 6,b,c$ satisfy matrix value equal to zero.
Then we will get
$\begin{align}
& \Rightarrow a{{b}^{2}}-{{a}^{2}}c-6{{b}^{2}}+6ac=0 \\
& \Rightarrow a{{b}^{2}}-6{{b}^{2}}-{{a}^{2}}c+6ac=0 \\
& \Rightarrow {{b}^{2}}\left( a-6 \right)-ac\left( a-6 \right)=0 \\
& \Rightarrow \left( a-6 \right)\left( {{b}^{2}}-ac \right)=0 \\
\end{align}$
If $a\ne 6$ then $\left( {{b}^{2}}-ac \right)=0$ then simplifying the obtained equation we will get
$\begin{align}
& \Rightarrow {{b}^{2}}-ac=0 \\
& \Rightarrow {{b}^{2}}=ac \\
& \Rightarrow abc={{b}^{3}} \\
\end{align}$
Hence we get the value of $abc$ as ${{b}^{3}}$.
Option C is the correct answer.
Note:
Students must remember the condition given in the question that $a\ne 6$. If we consider the factor $a-6=0$ then we will get the value $a=6$. So we need to ignore the factor. In matrices, determinants are the special numbers calculated from the square matrix. Square matrix should have an equal number of rows and columns.
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)