Advertisements
Advertisements
Question
Solve the system of linear equations using the matrix method.
2x + 3y + 3z = 5
x − 2y + z = −4
3x − y − 2z = 3
Advertisements
Solution
`[(2,3,3),(1,-2,1),(3,-1,-2)] [(x),(y),(z)] = [(5),(-4),(3)]` AX = B
A = `[(2,3,3),(1,-2,1),(3,-1,-2)]`, X = `[(x),(y),(z)]` or B = `[(5),(-4),(3)]`
Now, |A| = `|(2,3,3),(1,-2,1),(3,-1,-2)|`
= 2(4 + 1) − 3(−2 − 3) + 3(−1 + 6)
= 10 + 15 + 15
= 40 ≠ 0
∴ A−1 exists and hence the given equation has a unique solution.
A11 = `(-1)^(1 + 1) |(-2,1),(-1,-2)|`
= 4 + 1
= 5
A12 = `(-1)^(1 + 2) |(1,1),(3,-2)|`
= (−2 − 3)
= 5
A13 = `(-1)^(1 + 3) |(1,-2),(3,-1)|`
= −1 + 6
= 5
A21 = `(-1)^(2 + 1) |(3,3),(-1,-2)|`
= −(−6 + 3)
= 3
A22 = `(-1)^(2 + 2) |(2,3),(3,-2)|`
= −4 − 9
= −13
A23 = `(-1)^(2 + 3) |(2,3),(3,-1)|`
= −(−2 − 9)
= 11
A31 = `(-1)^ (3 + 1) |(3,3),(-2,1)|`
= 3 + 6
= 9
A32 = `(-1)^(3 + 2) |(2,3),(1,1)|`
= −(2 − 3)
= 1
A33 = `(-1)^(3 + 3) |(2,3),(1, -2)|`
= −4 − 3
= −7
∴ A−1 = `1/|A|` (Adj A)
= `1/40 [(5,5,5),(3,-13,11),(9,1,-7)]`
= `1/40 [(5,3,9),(5,-13,1),(5,11,-7)]`
X = A−1B
⇒ `[(x),(y),(z)] = 1/40 [(5,3,9),(5,-13,1),(5,11,-7)] [(5),(-4),(3)]`
= `1/40 [(25 - 12 + 27),(25 + 52 + 3),(25 - 44 - 21)]`
= `1/40 [(40),(80),(-40)]`
= `[(1),(2),(-1)]`
∴ x = 1, y = 2 and z = −1
APPEARS IN
RELATED QUESTIONS
Examine the consistency of the system of equations.
x + y + z = 1
2x + 3y + 2z = 2
ax + ay + 2az = 4
Solve the system of linear equations using the matrix method.
4x – 3y = 3
3x – 5y = 7
Evaluate the following determinant:
\[\begin{vmatrix}1 & 4 & 9 \\ 4 & 9 & 16 \\ 9 & 16 & 25\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}\cos\left( x + y \right) & - \sin\left( x + y \right) & \cos2y \\ \sin x & \cos x & \sin y \\ - \cos x & \sin x & - \cos y\end{vmatrix}\]
Evaluate :
\[\begin{vmatrix}a & b & c \\ c & a & b \\ b & c & a\end{vmatrix}\]
Using properties of determinants prove that
\[\begin{vmatrix}x + 4 & 2x & 2x \\ 2x & x + 4 & 2x \\ 2x & 2x & x + 4\end{vmatrix} = \left( 5x + 4 \right) \left( 4 - x \right)^2\]
\[\begin{vmatrix}1 + a & 1 & 1 \\ 1 & 1 + a & a \\ 1 & 1 & 1 + a\end{vmatrix} = a^3 + 3 a^2\]
Solve the following determinant equation:
If a, b, c are real numbers such that
\[\begin{vmatrix}b + c & c + a & a + b \\ c + a & a + b & b + c \\ a + b & b + c & c + a\end{vmatrix} = 0\] , then show that either
\[a + b + c = 0 \text{ or, } a = b = c\]
Using determinants show that the following points are collinear:
(5, 5), (−5, 1) and (10, 7)
Using determinants prove that the points (a, b), (a', b') and (a − a', b − b') are collinear if ab' = a'b.
Using determinants, find the area of the triangle with vertices (−3, 5), (3, −6), (7, 2).
x − 2y = 4
−3x + 5y = −7
Prove that :
Prove that :
3x − y + 2z = 3
2x + y + 3z = 5
x − 2y − z = 1
Solve each of the following system of homogeneous linear equations.
x + y − 2z = 0
2x + y − 3z = 0
5x + 4y − 9z = 0
Solve each of the following system of homogeneous linear equations.
3x + y + z = 0
x − 4y + 3z = 0
2x + 5y − 2z = 0
Find the value of the determinant
\[\begin{bmatrix}4200 & 4201 \\ 4205 & 4203\end{bmatrix}\]
Evaluate \[\begin{vmatrix}4785 & 4787 \\ 4789 & 4791\end{vmatrix}\]
If \[A = \left[ a_{ij} \right]\] is a 3 × 3 diagonal matrix such that a11 = 1, a22 = 2 a33 = 3, then find |A|.
Write the value of the determinant \[\begin{vmatrix}2 & - 3 & 5 \\ 4 & - 6 & 10 \\ 6 & - 9 & 15\end{vmatrix} .\]
Find the value of x from the following : \[\begin{vmatrix}x & 4 \\ 2 & 2x\end{vmatrix} = 0\]
If \[A = \begin{bmatrix}5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3\end{bmatrix}\]. Write the cofactor of the element a32.
If ω is a non-real cube root of unity and n is not a multiple of 3, then \[∆ = \begin{vmatrix}1 & \omega^n & \omega^{2n} \\ \omega^{2n} & 1 & \omega^n \\ \omega^n & \omega^{2n} & 1\end{vmatrix}\]
The maximum value of \[∆ = \begin{vmatrix}1 & 1 & 1 \\ 1 & 1 + \sin\theta & 1 \\ 1 + \cos\theta & 1 & 1\end{vmatrix}\] is (θ is real)
Solve the following system of equations by matrix method:
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
Show that each one of the following systems of linear equation is inconsistent:
4x − 2y = 3
6x − 3y = 5
Show that each one of the following systems of linear equation is inconsistent:
x + y − 2z = 5
x − 2y + z = −2
−2x + y + z = 4
Two schools P and Q want to award their selected students on the values of Discipline, Politeness and Punctuality. The school P wants to award ₹x each, ₹y each and ₹z each the three respectively values to its 3, 2 and 1 students with a total award money of ₹1,000. School Q wants to spend ₹1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for three values as before). If the total amount of awards for one prize on each value is ₹600, using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.
Solve the following for x and y: \[\begin{bmatrix}3 & - 4 \\ 9 & 2\end{bmatrix}\binom{x}{y} = \binom{10}{ 2}\]
Write the value of `|(a-b, b- c, c-a),(b-c, c-a, a-b),(c-a, a-b, b-c)|`
Solve the following equations by using inversion method.
x + y + z = −1, x − y + z = 2 and x + y − z = 3
Let `θ∈(0, π/2)`. If the system of linear equations,
(1 + cos2θ)x + sin2θy + 4sin3θz = 0
cos2θx + (1 + sin2θ)y + 4sin3θz = 0
cos2θx + sin2θy + (1 + 4sin3θ)z = 0
has a non-trivial solution, then the value of θ is
______.
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 ______.
