Advertisements
Advertisements
प्रश्न
x + y = 1
x + z = − 6
x − y − 2z = 3
Advertisements
उत्तर
These equations can be written as
x+ y + 0z = 1
x + 0y + z = − 6
x − y − 2z = 3
\[D = \begin{vmatrix}1 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & - 1 & - 2\end{vmatrix}\]
\[ = 1(0 + 1) - 1( - 2 - 1) + 0( - 1 - 0)\]
\[ = 4\]
\[ D_1 = \begin{vmatrix}1 & 1 & 0 \\ - 6 & 0 & 1 \\ 3 & - 1 & - 2\end{vmatrix}\]
\[ = 1(0 + 1) - 1(12 - 3) + 0(6 - 0)\]
\[ = - 8\]
\[ D_2 = \begin{vmatrix}1 & 1 & 0 \\ 1 & - 6 & 1 \\ 1 & 3 & - 2\end{vmatrix}\]
\[ = 1(12 - 3) - 1( - 2 - 1) + 0(3 + 6)\]
\[ = 12\]
\[ D_3 = \begin{vmatrix}1 & 1 & 1 \\ 1 & 0 & - 6 \\ 1 & - 1 & 3\end{vmatrix}\]
\[ = 1(0 - 6) - 1(3 + 6) + 1( - 1 - 0)\]
\[ = - 16\]
\[ \text{ Now } , \]
\[x = \frac{D_1}{D} = \frac{- 8}{4} = - 2\]
\[y = \frac{D_2}{D} = \frac{12}{4} = 3\]
\[z = \frac{D_3}{D} = \frac{- 16}{4} = - 4\]
\[ \therefore x = - 2, y = 3 \text{ and } z = - 4\]
APPEARS IN
संबंधित प्रश्न
Solve the system of linear equations using the matrix method.
5x + 2y = 4
7x + 3y = 5
Solve the system of linear equations using the matrix method.
2x – y = –2
3x + 4y = 3
Solve the system of linear equations using the matrix method.
5x + 2y = 3
3x + 2y = 5
Solve the system of linear equations using the matrix method.
x − y + z = 4
2x + y − 3z = 0
x + y + z = 2
Evaluate the following determinant:
\[\begin{vmatrix}\cos \theta & - \sin \theta \\ \sin \theta & \cos \theta\end{vmatrix}\]
Evaluate
\[\begin{vmatrix}2 & 3 & - 5 \\ 7 & 1 & - 2 \\ - 3 & 4 & 1\end{vmatrix}\] by two methods.
Find the value of x, if
\[\begin{vmatrix}2 & 4 \\ 5 & 1\end{vmatrix} = \begin{vmatrix}2x & 4 \\ 6 & x\end{vmatrix}\]
Find the value of x, if
\[\begin{vmatrix}x + 1 & x - 1 \\ x - 3 & x + 2\end{vmatrix} = \begin{vmatrix}4 & - 1 \\ 1 & 3\end{vmatrix}\]
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:
\[\begin{vmatrix}1^2 & 2^2 & 3^2 & 4^2 \\ 2^2 & 3^2 & 4^2 & 5^2 \\ 3^2 & 4^2 & 5^2 & 6^2 \\ 4^2 & 5^2 & 6^2 & 7^2\end{vmatrix}\]
Evaluate :
\[\begin{vmatrix}1 & a & bc \\ 1 & b & ca \\ 1 & c & ab\end{vmatrix}\]
Prove that
\[\begin{vmatrix}\frac{a^2 + b^2}{c} & c & c \\ a & \frac{b^2 + c^2}{a} & a \\ b & b & \frac{c^2 + a^2}{b}\end{vmatrix} = 4abc\]
Using properties of determinants prove that
\[\begin{vmatrix}x + 4 & 2x & 2x \\ 2x & x + 4 & 2x \\ 2x & 2x & x + 4\end{vmatrix} = \left( 5x + 4 \right) \left( 4 - x \right)^2\]
\[\begin{vmatrix}- a \left( b^2 + c^2 - a^2 \right) & 2 b^3 & 2 c^3 \\ 2 a^3 & - b \left( c^2 + a^2 - b^2 \right) & 2 c^3 \\ 2 a^3 & 2 b^3 & - c \left( a^2 + b^2 - c^2 \right)\end{vmatrix} = abc \left( a^2 + b^2 + c^2 \right)^3\]
Show that
Using determinants show that the following points are collinear:
(5, 5), (−5, 1) and (10, 7)
Prove that :
9x + 5y = 10
3y − 2x = 8
x − 4y − z = 11
2x − 5y + 2z = 39
− 3x + 2y + z = 1
2y − 3z = 0
x + 3y = − 4
3x + 4y = 3
3x − y + 2z = 6
2x − y + z = 2
3x + 6y + 5z = 20.
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.
If A is a singular matrix, then write the value of |A|.
Find the value of the determinant
\[\begin{bmatrix}101 & 102 & 103 \\ 104 & 105 & 106 \\ 107 & 108 & 109\end{bmatrix}\]
If the matrix \[\begin{bmatrix}5x & 2 \\ - 10 & 1\end{bmatrix}\] is singular, find the value of x.
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 \[\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & - 2 \\ 7 & 3\end{vmatrix}\] , write the value of x.
Find the maximum value of \[\begin{vmatrix}1 & 1 & 1 \\ 1 & 1 + \sin \theta & 1 \\ 1 & 1 & 1 + \cos \theta\end{vmatrix}\]
Using the factor theorem it is found that a + b, b + c and c + a are three factors of the determinant
The other factor in the value of the determinant is
Solve the following system of equations by matrix method:
5x + 2y = 3
3x + 2y = 5
Show that the following systems of linear equations is consistent and also find their solutions:
2x + 3y = 5
6x + 9y = 15
Show that each one of the following systems of linear equation is inconsistent:
2x + 5y = 7
6x + 15y = 13
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 − z = 0
x − 2y + z = 0
3x + 6y − 5z = 0
If ` abs((1 + "a"^2 "x", (1 + "b"^2)"x", (1 + "c"^2)"x"),((1 + "a"^2) "x", 1 + "b"^2 "x", (1 + "c"^2) "x"), ((1 + "a"^2) "x", (1 + "b"^2) "x", 1 + "c"^2 "x"))`, then f(x) is apolynomial of degree ____________.
`abs ((1, "a"^2 + "bc", "a"^3),(1, "b"^2 + "ca", "b"^3),(1, "c"^2 + "ab", "c"^3))`
If the system of equations x + ky - z = 0, 3x - ky - z = 0 & x - 3y + z = 0 has non-zero solution, then k is equal to ____________.
If A = `[(1,-1,0),(2,3,4),(0,1,2)]` and B = `[(2,2,-4),(-4,2,-4),(2,-1,5)]`, then:
What is the nature of the given system of equations
`{:(x + 2y = 2),(2x + 3y = 3):}`
Let A = `[(i, -i),(-i, i)], i = sqrt(-1)`. Then, the system of linear equations `A^8[(x),(y)] = [(8),(64)]` has ______.
