Advertisements
Advertisements
प्रश्न
If A = `[(1, -1, 2),(3, 0, -2),(1, 0, 3)]`, verify that A(adj A) = (adj A)A
Advertisements
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
Let A be a nonsingular square matrix of order 3 × 3. Then |adj A| is equal to ______.
Examine the consistency of the system of equations.
x + y + z = 1
2x + 3y + 2z = 2
ax + ay + 2az = 4
Examine the consistency of the system of equations.
5x − y + 4z = 5
2x + 3y + 5z = 2
5x − 2y + 6z = −1
If A = `[(2,-3,5),(3,2,-4),(1,1,-2)]` find A−1. Using A−1 solve the system of equations:
2x – 3y + 5z = 11
3x + 2y – 4z = –5
x + y – 2z = –3
Evaluate the following determinant:
\[\begin{vmatrix}a + ib & c + id \\ - c + id & a - ib\end{vmatrix}\]
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
\[\begin{vmatrix}2 & 3 \\ 4 & 5\end{vmatrix} = \begin{vmatrix}x & 3 \\ 2x & 5\end{vmatrix}\]
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:
\[\begin{vmatrix}\sin^2 23^\circ & \sin^2 67^\circ & \cos180^\circ \\ - \sin^2 67^\circ & - \sin^2 23^\circ & \cos^2 180^\circ \\ \cos180^\circ & \sin^2 23^\circ & \sin^2 67^\circ\end{vmatrix}\]
Evaluate the following:
\[\begin{vmatrix}x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x\end{vmatrix}\]
Evaluate the following:
\[\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix}\]
\[\begin{vmatrix}0 & b^2 a & c^2 a \\ a^2 b & 0 & c^2 b \\ a^2 c & b^2 c & 0\end{vmatrix} = 2 a^3 b^3 c^3\]
Prove that
\[\begin{vmatrix}- bc & b^2 + bc & c^2 + bc \\ a^2 + ac & - ac & c^2 + ac \\ a^2 + ab & b^2 + ab & - ab\end{vmatrix} = \left( ab + bc + ca \right)^3\]
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:
Solve the following determinant equation:
Solve the following determinant equation:
Solve the following determinant equation:
Solve the following determinant equation:
If \[a, b\] and c are all non-zero and
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 :
Prove that :
x − y + 3z = 6
x + 3y − 3z = − 4
5x + 3y + 3z = 10
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 prove that ab + bc + ca = abc.
If \[A = \begin{bmatrix}1 & 2 \\ 3 & - 1\end{bmatrix}\text{ and }B = \begin{bmatrix}1 & 0 \\ - 1 & 0\end{bmatrix}\] , find |AB|.
If \[A = \begin{bmatrix}1 & 2 \\ 3 & - 1\end{bmatrix}\text{ and B} = \begin{bmatrix}1 & - 4 \\ 3 & - 2\end{bmatrix},\text{ find }|AB|\]
If \[\begin{vmatrix}2x + 5 & 3 \\ 5x + 2 & 9\end{vmatrix} = 0\]
If x ∈ N and \[\begin{vmatrix}x + 3 & - 2 \\ - 3x & 2x\end{vmatrix}\] = 8, then find the value of x.
Let \[\begin{vmatrix}x & 2 & x \\ x^2 & x & 6 \\ x & x & 6\end{vmatrix} = a x^4 + b x^3 + c x^2 + dx + e\]
Then, the value of \[5a + 4b + 3c + 2d + e\] is equal to
If a, b, c are in A.P., then the determinant
\[\begin{vmatrix}x + 2 & x + 3 & x + 2a \\ x + 3 & x + 4 & x + 2b \\ x + 4 & x + 5 & x + 2c\end{vmatrix}\]
If \[\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & - 2 \\ 7 & 3\end{vmatrix}\] , then x =
The determinant \[\begin{vmatrix}b^2 - ab & b - c & bc - ac \\ ab - a^2 & a - b & b^2 - ab \\ bc - ca & c - a & ab - a^2\end{vmatrix}\]
Solve the following system of equations by matrix method:
5x + 3y + z = 16
2x + y + 3z = 19
x + 2y + 4z = 25
Solve the following system of equations by matrix method:
Show that each one of the following systems of linear equation is inconsistent:
4x − 2y = 3
6x − 3y = 5
2x − y + z = 0
3x + 2y − z = 0
x + 4y + 3z = 0
Solve the following for x and y: \[\begin{bmatrix}3 & - 4 \\ 9 & 2\end{bmatrix}\binom{x}{y} = \binom{10}{ 2}\]
If \[A = \begin{bmatrix}1 & - 2 & 0 \\ 2 & 1 & 3 \\ 0 & - 2 & 1\end{bmatrix}\] ,find A–1 and hence solve the system of equations x – 2y = 10, 2x + y + 3z = 8 and –2y + z = 7.
Find the inverse of the following matrix, using elementary transformations:
`A= [[2 , 3 , 1 ],[2 , 4 , 1],[3 , 7 ,2]]`
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 x − y + z = 4, x − 2y + 2z = 9 and 2x + y + 3z = 1.
If the system of equations x + λy + 2 = 0, λx + y – 2 = 0, λx + λy + 3 = 0 is consistent, then
What is the nature of the given system of equations
`{:(x + 2y = 2),(2x + 3y = 3):}`
