Advertisements
Advertisements
प्रश्न
If A \[\begin{bmatrix}1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4\end{bmatrix}\] , then show that |3 A| = 27 |A|.
उत्तर
\[A = \begin{bmatrix}1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4\end{bmatrix}\]
\[ \Rightarrow 3A = \begin{bmatrix}3 & 0 & 3 \\ 0 & 3 & 6 \\ 0 & 0 & 12\end{bmatrix} \left[\text{ Multiplying each element of A by 3 }\right]\]
\[ \Rightarrow \begin{vmatrix}3A\end{vmatrix} = \left( - 1 \right)^{1 + 1} 3\left( 36 - 0 \right) + \left( - 1 \right)^{1 + 2} 0\left( 0 - 0 \right) + \left( - 1 \right)^{1 + 3} 3\left( 0 - 0 \right) = 3\left( 36 - 0 \right) - 0\left( 0 - 0 \right) + 3\left( 0 - 0 \right) \left[\text{ Expanding along }R_1 \right]\]
\[ = 3 \times 36 = 108 . . . \left( 1 \right)\]
\[ \Rightarrow \begin{vmatrix}A\end{vmatrix} = \left( - 1 \right)^{1 + 1} 1\left( 4 - 0 \right) + \left( - 1 \right)^{1 + 2} 0\left( 0 - 0 \right) + \left( - 1 \right)^{1 + 3} 1\left( 0 - 0 \right) = 1\left( 4 - 0 \right) - 0\left( 0 - 0 \right) + 1\left( 0 - 0 \right) = 4 \left[\text{ Expanding along }R_1 \right]\]
\[ \Rightarrow 27\left| A \right| = 27 \times 4 = 108 . . . \left( 2 \right)\]
\[ \therefore \left| 3A \right| = 27 \left| A \right| \left[\text{ From eqs . (1) and (2) }\right]\]
APPEARS IN
संबंधित प्रश्न
Examine the consistency of the system of equations.
5x − y + 4z = 5
2x + 3y + 5z = 2
5x − 2y + 6z = −1
Show that x = 2 is a root of the equation
Solve the following determinant equation:
Find the area of the triangle with vertice at the point:
(−1, −8), (−2, −3) and (3, 2)
Find the area of the triangle with vertice at the point:
(0, 0), (6, 0) and (4, 3)
Using determinants show that the following points are collinear:
(3, −2), (8, 8) and (5, 2)
Prove that :
9x + 5y = 10
3y − 2x = 8
x+ y = 5
y + z = 3
x + z = 4
2x − 3y − 4z = 29
− 2x + 5y − z = − 15
3x − y + 5z = − 11
x − y + z = 3
2x + y − z = 2
− x − 2y + 2z = 1
An automobile company uses three types of steel S1, S2 and S3 for producing three types of cars C1, C2and C3. Steel requirements (in tons) for each type of cars are given below :
| Cars C1 |
C2 | C3 | |
| Steel S1 | 2 | 3 | 4 |
| S2 | 1 | 1 | 2 |
| S3 | 3 | 2 | 1 |
Using Cramer's rule, find the number of cars of each type which can be produced using 29, 13 and 16 tons of steel of three types respectively.
For what value of x, the following matrix is singular?
If \[A = \begin{bmatrix}1 & 2 \\ 3 & - 1\end{bmatrix}\text{ and B} = \begin{bmatrix}1 & - 4 \\ 3 & - 2\end{bmatrix},\text{ find }|AB|\]
If the matrix \[\begin{bmatrix}5x & 2 \\ - 10 & 1\end{bmatrix}\] is singular, find the value of x.
Let \[\begin{vmatrix}x & 2 & x \\ x^2 & x & 6 \\ x & x & 6\end{vmatrix} = a x^4 + b x^3 + c x^2 + dx + e\]
Then, the value of \[5a + 4b + 3c + 2d + e\] is equal to
The number of distinct real roots of \[\begin{vmatrix}cosec x & \sec x & \sec x \\ \sec x & cosec x & \sec x \\ \sec x & \sec x & cosec x\end{vmatrix} = 0\] lies in the interval
\[- \frac{\pi}{4} \leq x \leq \frac{\pi}{4}\]
If \[\begin{vmatrix}a & p & x \\ b & q & y \\ c & r & z\end{vmatrix} = 16\] , then the value of \[\begin{vmatrix}p + x & a + x & a + p \\ q + y & b + y & b + q \\ r + z & c + z & c + r\end{vmatrix}\] is
Solve the following system of equations by matrix method:
3x + 7y = 4
x + 2y = −1
Solve the following system of equations by matrix method:
x + y − z = 3
2x + 3y + z = 10
3x − y − 7z = 1
Solve the following system of equations by matrix method:
5x + 3y + z = 16
2x + y + 3z = 19
x + 2y + 4z = 25
Solve the following system of equations by matrix method:
8x + 4y + 3z = 18
2x + y +z = 5
x + 2y + z = 5
Show that each one of the following systems of linear equation is inconsistent:
3x − y − 2z = 2
2y − z = −1
3x − 5y = 3
The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method, find the number of awardees of each category. Apart from these values, namely, honesty, cooperation and supervision, suggest one more value which the management of the colony must include for awards.
x + y + z = 0
x − y − 5z = 0
x + 2y + 4z = 0
Consider the system of equations:
a1x + b1y + c1z = 0
a2x + b2y + c2z = 0
a3x + b3y + c3z = 0,
if \[\begin{vmatrix}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{vmatrix}\]= 0, then the system has
Let a, b, c be positive real numbers. The following system of equations in x, y and z
(a) no solution
(b) unique solution
(c) infinitely many solutions
(d) finitely many solutions
If `alpha, beta, gamma` are in A.P., then `abs (("x" - 3, "x" - 4, "x" - alpha),("x" - 2, "x" - 3, "x" - beta),("x" - 1, "x" - 2, "x" - gamma)) =` ____________.
Let A = `[(1,sin α,1),(-sin α,1,sin α),(-1,-sin α,1)]`, where 0 ≤ α ≤ 2π, then:
The number of values of k for which the linear equations 4x + ky + 2z = 0, kx + 4y + z = 0 and 2x + 2y + z = 0 possess a non-zero solution 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.
If c < 1 and the system of equations x + y – 1 = 0, 2x – y – c = 0 and – bx+ 3by – c = 0 is consistent, then the possible real values of b are
If `|(x + a, beta, y),(a, x + beta, y),(a, beta, x + y)|` = 0, then 'x' is equal to
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
If the system of linear equations
2x + y – z = 7
x – 3y + 2z = 1
x + 4y + δz = k, where δ, k ∈ R has infinitely many solutions, then δ + k is equal to ______.
The number of real values λ, such that the system of linear equations 2x – 3y + 5z = 9, x + 3y – z = –18 and 3x – y + (λ2 – |λ|z) = 16 has no solution, is ______.
