Advertisements
Advertisements
प्रश्न
If A = `[(1, -1, 2),(3, 0, -2),(1, 0, 3)]`, verify that A(adj A) = (adj A)A
उत्तर
A = `[(1, -1, 2),(3, 0, -2),(1, 0, 3)]`
A11 = (−1)1+1 M11 = `1|(0, -2),(0, 3)|` = 1(0 − 0) = 1 × 0 = 0
A12 = (−1)1+2 M12 = `1|(3, 0),(1, 0)|` = −1(9 + 2) = −11
A13 = (−1)1+3 M13 = `1|(3, 0)(1, 0)|` = 1(0 − 0) = 0
A21 = (−1)2+1 M21 = `-1|(-1, 2),(0, 3)|` = −1(−3 − 0) = 3
A22 = (−1)2+2 M22 = `1|(1, 2),(1, 3)|` = 1(3 − 2) = 1
A23 = (−1)2+3 M23 = `-1|(1, -1),(1, 0)|` = −1(0 + 1) = −1
A31 = (−1)3+1 M31 = `1|(1, -1),(1, 0)|` = 1(2 − 0) = 2
A32 = (−1)3+2 M32 = `-1|(1, 2),(3, -2)|` = −1(−2 − 6) = 8
A33 = (−1)3+3 M33 = `1|(1, -1),(3, 0)|` = 1(0 + 3) = 3
Hence, matrix of the co-factors is
`[("A"_11, "A"_12, "A"_13),("A"_21, "A"_22, "A"_23),("A"_31, "A"_32, "A"_33)] = [(0, -11, 0),(3, 1, -1),(2, 8, 3)]`
= `["A"_"ij"]_(3 xx 3)`
Now, adj A = `["A"_"ij"]_(3 xx 3)^"T"`
= `[(0, 3, 2),(-11, 1, 8),(0, -1, 3)]`
∴ A(adj A) = `[(1, -1, 2),(3, 0, -2),(1, 0, 3)] [(0,3, 2),(-11,1,8),(0, -1, 3)]`
= `[(0 + 11 + 0, 3 - 1 - 2, 2 - 8 + 6),(0 + 0 + 0, 9 + 0 + 2, 6 + 0 - 6),(0 + 0 + 0, 3 + 0 - 3, 2 + 0 + 9)]`
= `[(11, 0, 0),(0, 11, 0),(0, 0, 11)]` .......(i)
(adj A)A = `[(0, 3, 2),(-11, 1, 8),(0, -1, 3)] [(1, -1, 2),(3, 0, -2),(1, 0, 3)]`
= `[(0 + 9 + 2, 0 + 0 + 0, 0 - 6 + 6),(-11 + 3 + 8, 11 + 0 + 0, -22 - 2 + 24),(0 - 3 + 3, 0 - 0 + 0, 0 + 2 + 9)]`
= `[(11, 0, 0,(0, 11, 0),(0 0 11)]` .......(ii)
From equations (i) and (ii), we get
A(adj A) = (adj A)A
APPEARS IN
संबंधित प्रश्न
If `|[x+1,x-1],[x-3,x+2]|=|[4,-1],[1,3]|`, then write the value of x.
Solve system of linear equations, using matrix method.
5x + 2y = 4
7x + 3y = 5
Solve the system of linear equations using the matrix method.
x − y + z = 4
2x + y − 3z = 0
x + y + z = 2
For what value of x the matrix A is singular?
\[ A = \begin{bmatrix}1 + x & 7 \\ 3 - x & 8\end{bmatrix}\]
Evaluate the following determinant:
\[\begin{vmatrix}67 & 19 & 21 \\ 39 & 13 & 14 \\ 81 & 24 & 26\end{vmatrix}\]
Without expanding, show that the value of the following determinant is zero:
\[\begin{vmatrix}49 & 1 & 6 \\ 39 & 7 & 4 \\ 26 & 2 & 3\end{vmatrix}\]
Evaluate the following:
\[\begin{vmatrix}x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x\end{vmatrix}\]
Evaluate the following:
\[\begin{vmatrix}0 & x y^2 & x z^2 \\ x^2 y & 0 & y z^2 \\ x^2 z & z y^2 & 0\end{vmatrix}\]
Prove the following identity:
\[\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix} = a^2 \left( a + x + y + z \right)\]
Prove the following identity:
`|(a^3,2,a),(b^3,2,b),(c^3,2,c)| = 2(a-b) (b-c) (c-a) (a+b+c)`
Without expanding, prove that
\[\begin{vmatrix}a & b & c \\ x & y & z \\ p & q & r\end{vmatrix} = \begin{vmatrix}x & y & z \\ p & q & r \\ a & b & c\end{vmatrix} = \begin{vmatrix}y & b & q \\ x & a & p \\ z & c & r\end{vmatrix}\]
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
\[a + b + c = 0 \text{ or, } a = b = c\]
If the points (a, 0), (0, b) and (1, 1) are collinear, prove that a + b = ab.
Using determinants, find the equation of the line joining the points
(1, 2) and (3, 6)
Find values of k, if area of triangle is 4 square units whose vertices are
(k, 0), (4, 0), (0, 2)
Prove that :
Prove that :
2x − y = 17
3x + 5y = 6
5x + 7y = − 2
4x + 6y = − 3
x+ y = 5
y + z = 3
x + z = 4
x + y + z + 1 = 0
ax + by + cz + d = 0
a2x + b2y + x2z + d2 = 0
If \[A = \begin{bmatrix}0 & i \\ i & 1\end{bmatrix}\text{ and }B = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}\] , find the value of |A| + |B|.
Evaluate \[\begin{vmatrix}4785 & 4787 \\ 4789 & 4791\end{vmatrix}\]
If A = [aij] is a 3 × 3 scalar matrix such that a11 = 2, then write the value of |A|.
If I3 denotes identity matrix of order 3 × 3, write the value of its determinant.
If \[\begin{vmatrix}2x + 5 & 3 \\ 5x + 2 & 9\end{vmatrix} = 0\]
If \[\begin{vmatrix}x & \sin \theta & \cos \theta \\ - \sin \theta & - x & 1 \\ \cos \theta & 1 & x\end{vmatrix} = 8\] , write the value of x.
If ω is a non-real cube root of unity and n is not a multiple of 3, then \[∆ = \begin{vmatrix}1 & \omega^n & \omega^{2n} \\ \omega^{2n} & 1 & \omega^n \\ \omega^n & \omega^{2n} & 1\end{vmatrix}\]
Solve the following system of equations by matrix method:
8x + 4y + 3z = 18
2x + y +z = 5
x + 2y + z = 5
A company produces three products every day. Their production on a certain day is 45 tons. It is found that the production of third product exceeds the production of first product by 8 tons while the total production of first and third product is twice the production of second product. Determine the production level of each product using matrix method.
2x − y + 2z = 0
5x + 3y − z = 0
x + 5y − 5z = 0
The existence of the unique solution of the system of equations:
x + y + z = λ
5x − y + µz = 10
2x + 3y − z = 6
depends on
Prove that (A–1)′ = (A′)–1, where A is an invertible matrix.
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 ____________.
In system of equations, if inverse of matrix of coefficients A is multiplied by right side constant B vector then resultant will be?
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.
The system of simultaneous linear equations kx + 2y – z = 1, (k – 1)y – 2z = 2 and (k + 2)z = 3 have a unique solution if k equals:
If `|(x + a, beta, y),(a, x + beta, y),(a, beta, x + y)|` = 0, then 'x' is equal to
