Answer
Verified
421.8k+ views
Hint: In these types of questions you first have to see what elements of the matrix are the same. Then see what elements you can take common. Then apply some operations on the determinant like subtracting two rows, adding two columns etc. Simplify the determinant and then expand.
Complete step-by-step answer:
Det. \[\left| {\begin{array}{*{20}{c}}
a&a&x \\
m&m&m \\
b&x&b
\end{array}} \right| = 0\]
Here , we have one complete row of m.
∴ taking m common, we get
\[m\left| {\begin{array}{*{20}{c}}
a&a&x \\
1&1&1 \\
b&x&b
\end{array}} \right| = 0\]
Now\[{c_2} \to {c_2} - {c_1}\] and \[{c_3} \to {c_3} - {c_1}\]
\[m\left| {\begin{array}{*{20}{c}}
a&0&{x - a} \\
1&0&0 \\
b&{x - b}&0
\end{array}} \right| = 0\]
Now expand along row 2, we get
\[m\left[ { - 1\left| {\begin{array}{*{20}{c}}
0&{x - a} \\
{x - b}&0
\end{array}} \right| + 0 + 0} \right] = 0\]
\[m\left[ { - \left( { - \left( {x - a} \right)\left( {x - b} \right)} \right)} \right] = 0\]
\[m\left( {x - a} \right)\left( {x - b} \right) = 0\]
This means m=0, (x-a) = 0 and (x-b) = 0
∴ x can have two values x = a and x = b.
∴ we have two correct options ‘A’ and ‘B’.
Note: For simplifying the determinant, we have to make as many elements of the determinant zero as possible. While applying the operations we can multiply some scalar quantities with the rows and columns to make the elements zero. Then expand through the row or the columns whose maximum no. of elements are zero for easy calculations.
Complete step-by-step answer:
Det. \[\left| {\begin{array}{*{20}{c}}
a&a&x \\
m&m&m \\
b&x&b
\end{array}} \right| = 0\]
Here , we have one complete row of m.
∴ taking m common, we get
\[m\left| {\begin{array}{*{20}{c}}
a&a&x \\
1&1&1 \\
b&x&b
\end{array}} \right| = 0\]
Now\[{c_2} \to {c_2} - {c_1}\] and \[{c_3} \to {c_3} - {c_1}\]
\[m\left| {\begin{array}{*{20}{c}}
a&0&{x - a} \\
1&0&0 \\
b&{x - b}&0
\end{array}} \right| = 0\]
Now expand along row 2, we get
\[m\left[ { - 1\left| {\begin{array}{*{20}{c}}
0&{x - a} \\
{x - b}&0
\end{array}} \right| + 0 + 0} \right] = 0\]
\[m\left[ { - \left( { - \left( {x - a} \right)\left( {x - b} \right)} \right)} \right] = 0\]
\[m\left( {x - a} \right)\left( {x - b} \right) = 0\]
This means m=0, (x-a) = 0 and (x-b) = 0
∴ x can have two values x = a and x = b.
∴ we have two correct options ‘A’ and ‘B’.
Note: For simplifying the determinant, we have to make as many elements of the determinant zero as possible. While applying the operations we can multiply some scalar quantities with the rows and columns to make the elements zero. Then expand through the row or the columns whose maximum no. of elements are zero for easy calculations.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Using the following information to help you answer class 12 chemistry CBSE
Trending doubts
State the differences between manure and fertilize class 8 biology CBSE
Why are xylem and phloem called complex tissues aBoth class 11 biology CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
What would happen if plasma membrane ruptures or breaks class 11 biology CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
What precautions do you take while observing the nucleus class 11 biology CBSE
What would happen to the life of a cell if there was class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE