Advertisements
Advertisements
प्रश्न
Solve the following system of equations by matrix method:
Advertisements
उत्तर
\[\text{ Let }\frac{1}{x}\text{ be }a,\frac{1}{y}\text{ be } b \text{ and }\frac{1}{z}\text{ be }c.\]
Here,
\[A = \begin{bmatrix}2 & 3 & 10 \\ 4 & - 6 & 5 \\ 6 & 9 & - 20\end{bmatrix}\]
\[\left| A \right| = \begin{vmatrix}2 & 3 & 10 \\ 4 & - 6 & 5 \\ 6 & 9 & - 20\end{vmatrix}\]
\[ = 2\left( 120 - 45 \right) - 3\left( - 80 - 30 \right) + 10(36 + 36)\]
\[ = 150 + 330 + 720\]
\[ = 1200\]
\[ {\text{ Let }C}_{ij} {\text{ be the cofactors of the elements a }}_{ij}\text{ in }A\left[ a_{ij} \right].\text{ Then,}\]
\[ C_{11} = \left( - 1 \right)^{1 + 1} \begin{vmatrix}- 6 & 5 \\ 9 & - 20\end{vmatrix} = 75, C_{12} = \left( - 1 \right)^{1 + 2} \begin{vmatrix}4 & 5 \\ 6 & - 20\end{vmatrix} = 110, C_{13} = \left( - 1 \right)^{1 + 3} \begin{vmatrix}4 & - 6 \\ 6 & 9\end{vmatrix} = 72\]
\[ C_{21} = \left( - 1 \right)^{2 + 1} \begin{vmatrix}3 & 10 \\ 9 & - 20\end{vmatrix} = 150, C_{22} = \left( - 1 \right)^{2 + 2} \begin{vmatrix}2 & 10 \\ 6 & - 20\end{vmatrix} = - 100, C_{23} = \left( - 1 \right)^{2 + 3} \begin{vmatrix}2 & 3 \\ 6 & 9\end{vmatrix} = 0\]
\[ C_{31} = \left( - 1 \right)^{3 + 1} \begin{vmatrix}3 & 10 \\ - 6 & 5\end{vmatrix} = 75, C_{32} = \left( - 1 \right)^{3 + 2} \begin{vmatrix}2 & 10 \\ 4 & 5\end{vmatrix} = 30, C_{33} = \left( - 1 \right)^{3 + 3} \begin{vmatrix}2 & 3 \\ 4 & - 6\end{vmatrix} = - 24\]
\[adj A = \begin{bmatrix}75 & 110 & 72 \\ 150 & - 100 & 0 \\ 75 & 30 & - 24\end{bmatrix}^T \]
\[ = \begin{bmatrix}75 & 150 & 75 \\ 110 & - 100 & 30 \\ 72 & 0 & - 24\end{bmatrix}\]
\[ \Rightarrow A^{- 1} = \frac{1}{\left| A \right|}adj A\]
\[ = \frac{1}{1200} \begin{bmatrix}75 & 150 & 75 \\ 110 & - 100 & 30 \\ 72 & 0 & - 24\end{bmatrix}\]
\[X = A^{- 1} B\]
\[ \Rightarrow \begin{bmatrix}a \\ b \\ c\end{bmatrix} = \frac{1}{1200} \begin{bmatrix}75 & 150 & 75 \\ 110 & - 100 & 30 \\ 72 & 0 & - 24\end{bmatrix}\begin{bmatrix}4 \\ 1 \\ 2\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}a \\ b \\ c\end{bmatrix} = \frac{1}{1200}\begin{bmatrix}300 + 150 + 150 \\ 440 - 100 + 60 \\ 288 - 48\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}a \\ b \\ c\end{bmatrix} = \frac{1}{1200}\begin{bmatrix}600 \\ 400 \\ 240\end{bmatrix}\]
\[ \Rightarrow x = \frac{1}{a} = \frac{1200}{600}, y = \frac{1}{b} = \frac{1200}{400}\text{ and }z = \frac{1}{c} = \frac{1200}{240}\]
\[ \therefore x = 2, y = 3\text{ and }z = 5\]
APPEARS IN
संबंधित प्रश्न
Solve the system of linear equations using the matrix method.
2x + 3y + 3z = 5
x − 2y + z = −4
3x − y − 2z = 3
Solve the system of linear equations using the matrix method.
x − y + 2z = 7
3x + 4y − 5z = −5
2x − y + 3z = 12
Evaluate the following determinant:
\[\begin{vmatrix}a + ib & c + id \\ - c + id & a - ib\end{vmatrix}\]
Evaluate
\[\begin{vmatrix}2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12\end{vmatrix}^2 .\]
For what value of x the matrix A is singular?
\[A = \begin{bmatrix}x - 1 & 1 & 1 \\ 1 & x - 1 & 1 \\ 1 & 1 & x - 1\end{bmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}a & h & g \\ h & b & f \\ g & f & c\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3\end{vmatrix}\]
Evaluate :
\[\begin{vmatrix}a & b & c \\ c & a & b \\ b & c & a\end{vmatrix}\]
Prove that
\[\begin{vmatrix}- bc & b^2 + bc & c^2 + bc \\ a^2 + ac & - ac & c^2 + ac \\ a^2 + ab & b^2 + ab & - ab\end{vmatrix} = \left( ab + bc + ca \right)^3\]
\[\begin{vmatrix}- a \left( b^2 + c^2 - a^2 \right) & 2 b^3 & 2 c^3 \\ 2 a^3 & - b \left( c^2 + a^2 - b^2 \right) & 2 c^3 \\ 2 a^3 & 2 b^3 & - c \left( a^2 + b^2 - c^2 \right)\end{vmatrix} = abc \left( a^2 + b^2 + c^2 \right)^3\]
Prove the following identity:
\[\begin{vmatrix}2y & y - z - x & 2y \\ 2z & 2z & z - x - y \\ x - y - z & 2x & 2x\end{vmatrix} = \left( x + y + z \right)^3\]
Solve the following determinant equation:
If \[\begin{vmatrix}a & b - y & c - z \\ a - x & b & c - z \\ a - x & b - y & c\end{vmatrix} =\] 0, then using properties of determinants, find the value of \[\frac{a}{x} + \frac{b}{y} + \frac{c}{z}\] , where \[x, y, z \neq\] 0
Using determinants show that the following points are collinear:
(3, −2), (8, 8) and (5, 2)
Find the value of \[\lambda\] so that the points (1, −5), (−4, 5) and \[\lambda\] are collinear.
If the points (3, −2), (x, 2), (8, 8) are collinear, find x using determinant.
Prove that :
Prove that :
Prove that :
Prove that
x − y + z = 3
2x + y − z = 2
− x − 2y + 2z = 1
If A is a singular matrix, then write the value of |A|.
State whether the matrix
\[\begin{bmatrix}2 & 3 \\ 6 & 4\end{bmatrix}\] is singular or non-singular.
Write the value of
If x ∈ N and \[\begin{vmatrix}x + 3 & - 2 \\ - 3x & 2x\end{vmatrix}\] = 8, then find the value of x.
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:
3x + 4y + 7z = 14
2x − y + 3z = 4
x + 2y − 3z = 0
Show that the following systems of linear equations is consistent and also find their solutions:
2x + 2y − 2z = 1
4x + 4y − z = 2
6x + 6y + 2z = 3
Given \[A = \begin{bmatrix}2 & 2 & - 4 \\ - 4 & 2 & - 4 \\ 2 & - 1 & 5\end{bmatrix}, B = \begin{bmatrix}1 & - 1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2\end{bmatrix}\] , find BA and use this to solve the system of equations y + 2z = 7, x − y = 3, 2x + 3y + 4z = 17
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.
Two factories decided to award their employees for three values of (a) adaptable tonew techniques, (b) careful and alert in difficult situations and (c) keeping clam in tense situations, at the rate of ₹ x, ₹ y and ₹ z per person respectively. The first factory decided to honour respectively 2, 4 and 3 employees with a total prize money of ₹ 29000. The second factory decided to honour respectively 5, 2 and 3 employees with the prize money of ₹ 30500. If the three prizes per person together cost ₹ 9500, then
i) represent the above situation by matrix equation and form linear equation using matrix multiplication.
ii) Solve these equation by matrix method.
iii) Which values are reflected in the questions?
Let \[X = \begin{bmatrix}x_1 \\ x_2 \\ x_3\end{bmatrix}, A = \begin{bmatrix}1 & - 1 & 2 \\ 2 & 0 & 1 \\ 3 & 2 & 1\end{bmatrix}\text{ and }B = \begin{bmatrix}3 \\ 1 \\ 4\end{bmatrix}\] . If AX = B, then X is equal to
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 ______
Solve the following system of equations by using inversion method
x + y = 1, y + z = `5/3`, z + x = `4/3`
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 a, b, c are non-zeros, then the system of equations (α + a)x + αy + αz = 0, αx + (α + b)y + αz = 0, αx+ αy + (α + c)z = 0 has a non-trivial solution if
If `|(x + 1, x + 2, x + a),(x + 2, x + 3, x + b),(x + 3, x + 4, x + c)|` = 0, then a, b, care in
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 ______.
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 ______.
