2x + 3y − Z = 0 X − Y − 2z = 0 3x + Y + 3z = 0 - Mathematics

प्रश्न

2x + 3y − z = 0
x − y − 2z = 0
3x + y + 3z = 0

उत्तर

The given system of homogeneous equations can be written in matrix form as follows:
$[\begin{matrix} 2 & 3 & - 1 \\ 1 & - 1 & - 2 \\ 3 & 1 & 3 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$
AX = O
Here,
$A = [\begin{matrix} 2 & 3 & - 1 \\ 1 & - 1 & - 2 \\ 3 & 1 & 3 \end{matrix}], X = [\begin{matrix} x \\ y \\ z \end{matrix}] and O = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$
Now,
$| A | = | \begin{matrix} 2 & 3 & - 1 \\ 1 & - 1 & - 2 \\ 3 & 1 & 3 \end{matrix} |$
$= 2 (- 3 + 2) - 3 (3 + 6) - 1 (1 + 3)$
$= - 2 - 27 - 4$
$= - 33 \neq 0$
So, the given systemof homogeneous equations has only trivial solution, which is given below:
$x = y = z = 0$

shaalaa.com

Applications of Determinants and Matrices

क्या इस प्रश्न या उत्तर में कोई त्रुटि है?

Q 8 Q 7 Q 1

अध्याय 8: Solution of Simultaneous Linear Equations - Exercise 8.2 [पृष्ठ २१]

APPEARS IN

आरडी शर्मा Mathematics [English] Class 12

अध्याय 8 Solution of Simultaneous Linear Equations
Exercise 8.2 | Q 8 | पृष्ठ २१

वीडियो ट्यूटोरियलVIEW ALL [5]

view
वीडियो ट्यूटोरियल For All Subjects
Applications of Determinants and Matrices

video tutorial00:15:29
Applications of Determinants and Matrices

video tutorial00:27:00
Applications of Determinants and Matrices

video tutorial00:36:09
Applications of Determinants and Matrices

video tutorial00:11:34
Applications of Determinants and Matrices

video tutorial00:23:43

संबंधित प्रश्न

Let A be a nonsingular square matrix of order 3 × 3. Then |adj A| is equal to ______.

Find the value of x, if

$| \begin{matrix} 2 & 3 \\ 4 & 5 \end{matrix} | = | \begin{matrix} x & 3 \\ 2 x & 5 \end{matrix} |$

Evaluate the following determinant:

$| \begin{matrix} 67 & 19 & 21 \\ 39 & 13 & 14 \\ 81 & 24 & 26 \end{matrix} |$

Without expanding, show that the value of the following determinant is zero:

$| \begin{matrix} 0 & x & y \\ - x & 0 & z \\ - y & - z & 0 \end{matrix} |$

Evaluate the following:

$| \begin{matrix} a + x & y & z \\ x & a + y & z \\ x & y & a + z \end{matrix} |$

Prove that:

$[\begin{matrix} a & b & c \\ a - b & b - c & c - a \\ b + c & c + a & a + b \end{matrix}] = a^{3} + b^{3} + c^{3} - 3 a b c$

$| \begin{matrix} 0 & b^{2} a & c^{2} a \\ a^{2} b & 0 & c^{2} b \\ a^{2} c & b^{2} c & 0 \end{matrix} | = 2 a^{3} b^{3} c^{3}$

Show that x = 2 is a root of the equation

| \begin{matrix} x & - 6 & - 1 \\ 2 & - 3 x & x - 3 \\ - 3 & 2 x & x + 2 \end{matrix} | = 0

and solve it completely.

Find the area of the triangle with vertice at the point:

(−1, −8), (−2, −3) and (3, 2)

Find values of k, if area of triangle is 4 square units whose vertices are

(−2, 0), (0, 4), (0, k)

Prove that :

| \begin{matrix} 1 & a^{2} + b c & a^{3} \\ 1 & b^{2} + c a & b^{3} \\ 1 & c^{2} + a b & c^{3} \end{matrix} | = - (a - b) (b - c) (c - a) (a^{2} + b^{2} + c^{2})

Prove that :

| \begin{matrix} x + 4 & x & x \\ x & x + 4 & x \\ x & x & x + 4 \end{matrix} | = 16 (3 x + 4)

| \begin{matrix} 1 & a & a^{2} \\ a^{2} & 1 & a \\ a & a^{2} & 1 \end{matrix} | = {(a^{3} - 1)}^{2}

6x + y − 3z = 5
x + 3y − 2z = 5
2x + y + 4z = 8

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.

Solve each of the following system of homogeneous linear equations.
2x + 3y + 4z = 0
x + y + z = 0
2x − y + 3z = 0

Evaluate $| \begin{matrix} 4785 & 4787 \\ 4789 & 4791 \end{matrix} |$

If w is an imaginary cube root of unity, find the value of $| \begin{matrix} 1 & w & w^{2} \\ w & w^{2} & 1 \\ w^{2} & 1 & w \end{matrix} |$

If I₃ denotes identity matrix of order 3 × 3, write the value of its determinant.

Find the value of the determinant $| \begin{matrix} 2^{2} & 2^{3} & 2^{4} \\ 2^{3} & 2^{4} & 2^{5} \\ 2^{4} & 2^{5} & 2^{6} \end{matrix} |$ .

Write the cofactor of a₁₂ in the following matrix $[\begin{matrix} 2 & - 3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & - 7 \end{matrix}] .$

Find the maximum value of $| \begin{matrix} 1 & 1 & 1 \\ 1 & 1 + \sin θ & 1 \\ 1 & 1 & 1 + \cos θ \end{matrix} |$

Let $| \begin{matrix} x & 2 & x \\ x^{2} & x & 6 \\ x & x & 6 \end{matrix} | = a x^{4} + b x^{3} + c x^{2} + d x + e$
Then, the value of $5 a + 4 b + 3 c + 2 d + e$ is equal to

If $∆_{1} = | \begin{matrix} 1 & 1 & 1 \\ a & b & c \\ a^{2} & b^{2} & c^{2} \end{matrix} |, ∆_{2} = | \begin{matrix} 1 & b c & a \\ 1 & c a & b \\ 1 & a b & c \end{matrix} |, then$ }

Solve the following system of equations by matrix method:
3x + 4y − 5 = 0
x − y + 3 = 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 + 2z = 7
3x + 4y − 5z = −5
2x − y + 3z = 12

Show that each one of the following systems of linear equation is inconsistent:
3x − y − 2z = 2
2y − z = −1
3x − 5y = 3

If $A = [\begin{matrix} 2 & 3 & 1 \\ 1 & 2 & 2 \\ 3 & 1 & - 1 \end{matrix}]$ , find A^–1 and hence solve the system of equations 2x + y – 3z = 13, 3x + 2y + z = 4, x + 2y – z = 8.

Two institutions decided to award their employees for the three values of resourcefulness, competence and determination in the form of prices at the rate of Rs. x, y and z respectively per person. The first institution decided to award respectively 4, 3 and 2 employees with a total price money of Rs. 37000 and the second institution decided to award respectively 5, 3 and 4 employees with a total price money of Rs. 47000. If all the three prices per person together amount to Rs. 12000 then using matrix method find the value of x, y and z. What values are described in this equations?

If $[\begin{matrix} 1 & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} x \\ - 1 \\ z \end{matrix}] = [\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}]$ , find x, y and z.

Show that $| \begin{matrix} y + z & x & y \\ z + x & z & x \\ x + y & y & z \end{matrix} | = (x + y + z) {(x - z)}^{2}$

Transform $[\begin{matrix} 1 & 2 & 4 \\ 3 & - 1 & 5 \\ 2 & 4 & 6 \end{matrix}]$ into an upper triangular matrix by using suitable row transformations

If A = $[\begin{matrix} 1 & - 1 & 2 \\ 3 & 0 & - 2 \\ 1 & 0 & 3 \end{matrix}]$ , verify that A(adj A) = (adj A)A

If A = $[\begin{matrix} 1 & - 1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{matrix}]$ and B = $[\begin{matrix} 2 & 2 & - 4 \\ - 4 & 2 & - 4 \\ 2 & - 1 & 5 \end{matrix}]$ , then:

In system of equations, if inverse of matrix of coefficients A is multiplied by right side constant B vector then resultant will be?

If the system of equations x + λy + 2 = 0, λx + y – 2 = 0, λx + λy + 3 = 0 is consistent, then

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 ______.

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 ab + bc + ca equals ______.

2x + 3y − Z = 0 X − Y − 2z = 0 3x + Y + 3z = 0 - Mathematics

Advertisements

Advertisements

प्रश्न

उत्तर

APPEARS IN

वीडियो ट्यूटोरियलVIEW ALL [5]

संबंधित प्रश्न

Select a course

2x + 3y − Z = 0 X − Y − 2z = 0 3x + Y + 3z = 0 - Mathematics

Advertisements

Advertisements

प्रश्न

उत्तरShow Solution

APPEARS IN

वीडियो ट्यूटोरियलVIEW ALL [5]

संबंधित प्रश्न

Select a course

उत्तर