Advertisements
Advertisements
प्रश्न
Solve the following determinant equation:
Advertisements
उत्तर
\[\text{ Let }∆ = \begin{vmatrix}x + a & b & c \\ a & x + b & c \\ a & b & x + c\end{vmatrix}\]
\[ = \begin{vmatrix}x + a + b + c & b & c \\ x + a + b + c & x + b & c \\ x + a + b + c & b & x + c\end{vmatrix} \left[\text{ Applying }C_1 \text{ to }C_1 + C_2 + C_3 \right]\]
\[ = \left( x + a + b + c \right)\begin{vmatrix}1 & b & c \\ 1 & x + b & c \\ 1 & b & x + c\end{vmatrix} \]
\[ = \left( x + a + b + c \right)\begin{vmatrix}1 & b & c \\ 0 & x & 0 \\ 1 & b & x + c\end{vmatrix} \left[\text{ Applying }R_2 \text{ to }R_2 - R_1 \right]\]
\[ = \left( x + a + b + c \right)\begin{vmatrix}1 & b & c \\ 0 & x & 0 \\ 0 & 0 & x\end{vmatrix} \left[\text{ Applying }R_3 \text{ to }R_3 - R_1 \right]\]
\[ ∆ = \left( x + a + b + c \right)\left( x^2 - 0 \right) = 0 \left[\text{ Given }\right]\]
\[ \Rightarrow x^2 = 0\text{ or }x + a + b + c = 0\]
\[ \Rightarrow x = 0\text{ or }x = - \left( a + b + c \right)\]
APPEARS IN
संबंधित प्रश्न
Examine the consistency of the system of equations.
x + 2y = 2
2x + 3y = 3
If A \[\begin{bmatrix}1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4\end{bmatrix}\] , then show that |3 A| = 27 |A|.
Find the value of x, if
\[\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & 5 \\ 8 & 3\end{vmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}6 & - 3 & 2 \\ 2 & - 1 & 2 \\ - 10 & 5 & 2\end{vmatrix}\]
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:
\[\begin{vmatrix}1/a & a^2 & bc \\ 1/b & b^2 & ac \\ 1/c & c^2 & ab\end{vmatrix}\]
Evaluate :
\[\begin{vmatrix}1 & a & bc \\ 1 & b & ca \\ 1 & c & ab\end{vmatrix}\]
\[\begin{vmatrix}b^2 + c^2 & ab & ac \\ ba & c^2 + a^2 & bc \\ ca & cb & a^2 + b^2\end{vmatrix} = 4 a^2 b^2 c^2\]
Solve the following determinant equation:
Solve the following determinant equation:
Using determinants, find the equation of the line joining the points
(3, 1) and (9, 3)
Find values of k, if area of triangle is 4 square units whose vertices are
(k, 0), (4, 0), (0, 2)
x − 2y = 4
−3x + 5y = −7
Prove that :
Prove that :
Prove that
3x + y + z = 2
2x − 4y + 3z = − 1
4x + y − 3z = − 11
x + 2y = 5
3x + 6y = 15
If a, b, c are non-zero real numbers and if the system of equations
(a − 1) x = y + z
(b − 1) y = z + x
(c − 1) z = x + y
has a non-trivial solution, then prove that ab + bc + ca = abc.
Find the value of the determinant
\[\begin{bmatrix}4200 & 4201 \\ 4205 & 4203\end{bmatrix}\]
If A = [aij] is a 3 × 3 scalar matrix such that a11 = 2, then write the value of |A|.
Find the value of x from the following : \[\begin{vmatrix}x & 4 \\ 2 & 2x\end{vmatrix} = 0\]
Write the value of the determinant \[\begin{vmatrix}2 & 3 & 4 \\ 5 & 6 & 8 \\ 6x & 9x & 12x\end{vmatrix}\]
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:
6x − 12y + 25z = 4
4x + 15y − 20z = 3
2x + 18y + 15z = 10
Solve the following system of equations by matrix method:
3x + 4y + 2z = 8
2y − 3z = 3
x − 2y + 6z = −2
Solve the following system of equations by matrix method:
2x + y + z = 2
x + 3y − z = 5
3x + y − 2z = 6
Show that the following systems of linear equations is consistent and also find their solutions:
5x + 3y + 7z = 4
3x + 26y + 2z = 9
7x + 2y + 10z = 5
Use product \[\begin{bmatrix}1 & - 1 & 2 \\ 0 & 2 & - 3 \\ 3 & - 2 & 4\end{bmatrix}\begin{bmatrix}- 2 & 0 & 1 \\ 9 & 2 & - 3 \\ 6 & 1 & - 2\end{bmatrix}\] to solve the system of equations x + 3z = 9, −x + 2y − 2z = 4, 2x − 3y + 4z = −3.
The sum of three numbers is 2. If twice the second number is added to the sum of first and third, the sum is 1. By adding second and third number to five times the first number, we get 6. Find the three numbers by using matrices.
A shopkeeper has 3 varieties of pens 'A', 'B' and 'C'. Meenu purchased 1 pen of each variety for a total of Rs 21. Jeevan purchased 4 pens of 'A' variety 3 pens of 'B' variety and 2 pens of 'C' variety for Rs 60. While Shikha purchased 6 pens of 'A' variety, 2 pens of 'B' variety and 3 pens of 'C' variety for Rs 70. Using matrix method, find cost of each variety of pen.
The value of x, y, z for the following system of equations x + y + z = 6, x − y+ 2z = 5, 2x + y − z = 1 are ______
If A = `[(1, -1, 2),(3, 0, -2),(1, 0, 3)]`, verify that A(adj A) = (adj A)A
If `|(2x, 5),(8, x)| = |(6, -2),(7, 3)|`, then value of x is ______.
Choose the correct option:
If a, b, c are in A.P. then the determinant `[(x + 2, x + 3, x + 2a),(x + 3, x + 4, x + 2b),(x + 4, x + 5, x + 2c)]` is
The number of real value of 'x satisfying `|(x, 3x + 2, 2x - 1),(2x - 1, 4x, 3x + 1),(7x - 2, 17x + 6, 12x - 1)|` = 0 is
Let the system of linear equations x + y + az = 2; 3x + y + z = 4; x + 2z = 1 have a unique solution (x*, y*, z*). If (α, x*), (y*, α) and (x*, –y*) are collinear points, then the sum of absolute values of all possible values of α is ______.
