Advertisements
Advertisements
प्रश्न
उत्तर
\[\text{ Let LHS }= \Delta = \begin{vmatrix} 1 & a & a^2 \\ a^2 & 1 & a\\a & a^2 & 1 \end{vmatrix}\]
\[\Delta = \begin{vmatrix} 1 + a^2 + a & 1 + a^2 + a & 1 + a^2 + a\\ a^2 & 1 & a\\a & a^2 & 1 \end{vmatrix} \left[\text{ Applyng }R_1 \to R_1 + R_2 + R_2 \right]\]
\[ = \left( 1 + a^2 + a \right) \begin{vmatrix} 1 & 1 & 1 \\ a^2 & 1 & a\\a & a^2 & 1 \end{vmatrix} \left[\text{ Applying }C_2 \to C_2 - C_1\text{ and }C_3 \to C_3 - C_1 \right]\]
\[ = \left( 1 + a^2 + a \right) \begin{vmatrix} 1 & 0 & 0 \\ a^2 & 1 - a^2 & a - a^2 \\a & a^2 - a & 1 - a \end{vmatrix}\]
\[ = \left( 1 + a^2 + a \right) \begin{vmatrix} 1 & 0 & 0 \\ a^2 & \left( 1 - a \right)\left( 1 + a \right) & a\left( 1 - a \right)\\a & a\left( a - 1 \right) & 1 - a \end{vmatrix}\]
\[ = \left( 1 + a^2 + a \right)\left( a - 1 \right)\left( a - 1 \right) \begin{vmatrix} 1 & 0 & 0\\ a^2 & - \left( 1 + a \right) & - a\\a & a & - 1 \end{vmatrix} \left[\text{ Taking out (a - 1) common from }C_2\text{ and }C_3 \right]\]
\[ = \left( a^3 - 1 \right)\left\{ \left( a - 1 \right) \begin{vmatrix} 1 & 0 & 0\\a & - \left( 1 + a \right) & - a\\a & a & - 1 \end{vmatrix} \right\} \left[ \because \left( 1 + a^2 + a \right)\left( a - 1 \right) = \left( a^3 - 1 \right) \right]\]
\[ = \left( a^3 - 1 \right)\left\{ \left( a - 1 \right)\left( 1 + a^{} + a^2 \right) \right\}\]
\[ = \left( a^3 - 1 \right)\left( a^3 - 1 \right)\]
\[ = \left( a^3 - 1 \right)^2 \]
\[ = RHS \]
Hence proved.
APPEARS IN
संबंधित प्रश्न
Examine the consistency of the system of equations.
x + 3y = 5
2x + 6y = 8
Examine the consistency of the system of equations.
3x − y − 2z = 2
2y − z = −1
3x − 5y = 3
Solve system of linear equations, using matrix method.
5x + 2y = 3
3x + 2y = 5
Find the integral value of x, if \[\begin{vmatrix}x^2 & x & 1 \\ 0 & 2 & 1 \\ 3 & 1 & 4\end{vmatrix} = 28 .\]
For what value of x the matrix A is singular?
\[ A = \begin{bmatrix}1 + x & 7 \\ 3 - x & 8\end{bmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}0 & x & y \\ - x & 0 & z \\ - y & - z & 0\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}1^2 & 2^2 & 3^2 & 4^2 \\ 2^2 & 3^2 & 4^2 & 5^2 \\ 3^2 & 4^2 & 5^2 & 6^2 \\ 4^2 & 5^2 & 6^2 & 7^2\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\left( 2^x + 2^{- x} \right)^2 & \left( 2^x - 2^{- x} \right)^2 & 1 \\ \left( 3^x + 3^{- x} \right)^2 & \left( 3^x - 3^{- x} \right)^2 & 1 \\ \left( 4^x + 4^{- x} \right)^2 & \left( 4^x - 4^{- x} \right)^2 & 1\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\cos\left( x + y \right) & - \sin\left( x + y \right) & \cos2y \\ \sin x & \cos x & \sin y \\ - \cos x & \sin x & - \cos y\end{vmatrix}\]
2x − y = 1
7x − 2y = −7
Prove that :
Prove that
x + 2y = 5
3x + 6y = 15
A salesman has the following record of sales during three months for three items A, B and C which have different rates of commission
| Month | Sale of units | Total commission drawn (in Rs) |
||
| A | B | C | ||
| Jan | 90 | 100 | 20 | 800 |
| Feb | 130 | 50 | 40 | 900 |
| March | 60 | 100 | 30 | 850 |
Find out the rates of commission on items A, B and C by using determinant method.
Write the value of the determinant
\[\begin{bmatrix}2 & 3 & 4 \\ 2x & 3x & 4x \\ 5 & 6 & 8\end{bmatrix} .\]
Find the value of the determinant
\[\begin{bmatrix}101 & 102 & 103 \\ 104 & 105 & 106 \\ 107 & 108 & 109\end{bmatrix}\]
Find the value of the determinant \[\begin{vmatrix}243 & 156 & 300 \\ 81 & 52 & 100 \\ - 3 & 0 & 4\end{vmatrix} .\]
For what value of x is the matrix \[\begin{bmatrix}6 - x & 4 \\ 3 - x & 1\end{bmatrix}\] singular?
If \[\begin{vmatrix}2x & x + 3 \\ 2\left( x + 1 \right) & x + 1\end{vmatrix} = \begin{vmatrix}1 & 5 \\ 3 & 3\end{vmatrix}\], then write the value of x.
If \[\begin{vmatrix}3x & 7 \\ - 2 & 4\end{vmatrix} = \begin{vmatrix}8 & 7 \\ 6 & 4\end{vmatrix}\] , find the value of x.
If\[f\left( x \right) = \begin{vmatrix}0 & x - a & x - b \\ x + a & 0 & x - c \\ x + b & x + c & 0\end{vmatrix}\]
If \[x, y \in \mathbb{R}\], then the determinant
There are two values of a which makes the determinant \[∆ = \begin{vmatrix}1 & - 2 & 5 \\ 2 & a & - 1 \\ 0 & 4 & 2a\end{vmatrix}\] equal to 86. The sum of these two values is
Solve the following system of equations by matrix method:
5x + 2y = 3
3x + 2y = 5
Solve the following system of equations by matrix method:
3x + 4y + 7z = 14
2x − y + 3z = 4
x + 2y − 3z = 0
Solve the following system of equations by matrix method:
\[\frac{2}{x} - \frac{3}{y} + \frac{3}{z} = 10\]
\[\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 10\]
\[\frac{3}{x} - \frac{1}{y} + \frac{2}{z} = 13\]
Solve the following system of equations by matrix method:
x − y + z = 2
2x − y = 0
2y − z = 1
Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. The school A wants to award ₹x each, ₹y each and ₹z each for the three respective values to 3, 2 and 1 students respectively with a total award money of ₹1,600. School B wants to spend ₹2,300 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is ₹900, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award.
x + y − 6z = 0
x − y + 2z = 0
−3x + y + 2z = 0
If A = `[[1,1,1],[0,1,3],[1,-2,1]]` , find A-1Hence, solve the system of equations:
x +y + z = 6
y + 3z = 11
and x -2y +z = 0
Solve the following equations by using inversion method.
x + y + z = −1, x − y + z = 2 and x + y − z = 3
The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is ₹ 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is ₹ 90. Whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is ₹ 70. Find the cost of each item per dozen by using matrices
If `|(2x, 5),(8, x)| = |(6, -2),(7, 3)|`, then value of x is ______.
The value (s) of m does the system of equations 3x + my = m and 2x – 5y = 20 has a solution satisfying the conditions x > 0, y > 0.
The greatest value of c ε R for which the system of linear equations, x – cy – cz = 0, cx – y + cz = 0, cx + cy – z = 0 has a non-trivial solution, is ______.
