Advertisements
Advertisements
प्रश्न
If x, y, z are nonzero real numbers, then the inverse of matrix A = `[(x,0,0),(0,y,0),(0,0,z)]` is ______.
विकल्प
`[(x^(-1),0,0),(0, y^(-1),0),(0,0,z^(-1))]`
`xyz[(x^(-1),0,0),(0,y^(-1),0),(0,0,z^(-1))]`
`1/(xyz)[(x,0,0),(0,y,0),(0,0,z)]`
`1/(xyz)[(1,0,0),(0,1,0),(0,0,1)]`
Advertisements
उत्तर
If x, y, z are nonzero real numbers, then the inverse of matrix A = `[(x,0,0),(0,y,0),(0,0,z)]` is `underlinebb([(x^(-1),0,0),(0, y^(-1),0),(0,0,z^(-1))])`.
Explanation:
Let, A = `[(x,0,0),(0,y,0),(0,0,z)]`
∴ |A| = x[yz − 0] = xyz
∴ A11 = `(-1)^(1 + 1)[(y,0),(0,z)]`
= (−1)2 [yz − 0]
= 1 × yz
= yz
A12 = `(-1)^(1 + 2)[(0,0),(0,z)]`
= (−1)3 [0 − 0]
= 0
A13 = `(-1)^(1 + 3)[(0,y),(0,0)]`
= (−1)4 [0 − 0]
= 0
A21 = `(-1)^(2 + 1)[(0,0),(0,z)]`
= (−1)3 [0 − 0]
= 0
A22 = `(-1)^(2 + 2)[(x,0),(0,z)]`
= (−1)4 [xz − 0]
= 1 × zx
= zx
A23 = `(-1)^(2 + 3)[(x,0),(0,0)]`
= (−1)5 [0 − 0]
= 0
A31 = `(-1)^(3 + 1)[(0,0),(0,z)]`
= (−1)4 [0 − 0]
= 0
A32 = `(-1)^(3 + 2)[(x,0),(0,0)]`
= (−1)5 [0 − 0]
= 0
A33 = `(-1)^(3 + 3)[(x,0),(0,y)]`
= (−1)6 [xy − 0]
= xy
∴ adj A = `[(yz,0,0),(0,zx,0),(0,0,xy)] = [(yz,0,0),(0,zx,0),(0,0,xy)]`
A−1 = `1/|A|` (adj A)
= `1/(xyz)[(yz,0,0),(0,zx,0),(0,0,xy)]`
= `[(1/x,0,0),(0,1/y,0),(0,0,1/z)]`
= `[(x^-1,0,0),(0,y^-1,0),(0,0,z^-1)]`
APPEARS IN
संबंधित प्रश्न
Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. School A wants to award Rs x each, Rs y each and Rs z each for the three respective values to 3, 2 and 1 students, respectively with a total award money of Rs 1,600. School B wants to spend Rs 2,300 to award 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is Rs 900, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for an award.
Find the adjoint of the matrices.
`[(1,2),(3,4)]`
Find the inverse of the matrices (if it exists).
`[(-1,5),(-3,2)]`
Find the inverse of the matrices (if it exists).
`[(1,2,3),(0,2,4),(0,0,5)]`
Find the inverse of the matrices (if it exists).
`[(1,0,0),(3,3,0),(5,2,-1)]`
If A = `[(3,1),(-1,2)]` show that A2 – 5A + 7I = 0. Hence, find A–1.
If A−1 = `[(3,-1,1),(-15,6,-5),(5,-2,2)]` and B = `[(1,2,-2),(-1,3,0),(0,-2,1)]`, find (AB)−1.
Let A = `[(1,2,1),(2,3,1),(1,1,5)]` verify that
- [adj A]–1 = adj(A–1)
- (A–1)–1 = A
Let A = `[(1, sin theta, 1),(-sin theta,1,sin theta),(-1, -sin theta, 1)]` where 0 ≤ θ ≤ 2π, then ______.
Find the adjoint of the following matrix:
\[\begin{bmatrix}a & b \\ c & d\end{bmatrix}\]
Find the adjoint of the following matrix:
Verify that (adj A) A = |A| I = A (adj A) for the above matrix.
Compute the adjoint of the following matrix:
Verify that (adj A) A = |A| I = A (adj A) for the above matrix.
Find the inverse of the following matrix:
Find the inverse of the following matrix.
\[\begin{bmatrix}1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2\end{bmatrix}\]
Find the inverse of the following matrix.
Find the inverse of the following matrix.
Find the inverse of the following matrix and verify that \[A^{- 1} A = I_3\]
Find the inverse of the matrix \[A = \begin{bmatrix}a & b \\ c & \frac{1 + bc}{a}\end{bmatrix}\] and show that \[a A^{- 1} = \left( a^2 + bc + 1 \right) I - aA .\]
If \[A = \begin{bmatrix}4 & 3 \\ 2 & 5\end{bmatrix}\], find x and y such that
Show that \[A = \begin{bmatrix}5 & 3 \\ - 1 & - 2\end{bmatrix}\] satisfies the equation \[x^2 - 3x - 7 = 0\]. Thus, find A−1.
Show that \[A = \begin{bmatrix}6 & 5 \\ 7 & 6\end{bmatrix}\] satisfies the equation \[x^2 - 12x + 1 = O\]. Thus, find A−1.
If \[A = \begin{bmatrix}- 1 & 2 & 0 \\ - 1 & 1 & 1 \\ 0 & 1 & 0\end{bmatrix}\] , show that \[A^2 = A^{- 1} .\]
Find the matrix X satisfying the matrix equation \[X\begin{bmatrix}5 & 3 \\ - 1 & - 2\end{bmatrix} = \begin{bmatrix}14 & 7 \\ 7 & 7\end{bmatrix}\]
If \[A = \begin{bmatrix}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{bmatrix}\] , find \[A^{- 1}\] and prove that \[A^2 - 4A - 5I = O\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}2 & - 1 & 3 \\ 1 & 2 & 4 \\ 3 & 1 & 1\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}- 1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix}\]
If adj \[A = \begin{bmatrix}2 & 3 \\ 4 & - 1\end{bmatrix}\text{ and adj }B = \begin{bmatrix}1 & - 2 \\ - 3 & 1\end{bmatrix}\]
If A is a square matrix, then write the matrix adj (AT) − (adj A)T.
If A is an invertible matrix such that |A−1| = 2, find the value of |A|.
If A is an invertible matrix of order 3, then which of the following is not true ?
If \[A = \begin{bmatrix}3 & 4 \\ 2 & 4\end{bmatrix}, B = \begin{bmatrix}- 2 & - 2 \\ 0 & - 1\end{bmatrix},\text{ then }\left( A + B \right)^{- 1} =\]
If A, B are two n × n non-singular matrices, then __________ .
An amount of Rs 10,000 is put into three investments at the rate of 10, 12 and 15% per annum. The combined income is Rs 1310 and the combined income of first and second investment is Rs 190 short of the income from the third. Find the investment in each using matrix method.
Using matrix method, solve the following system of equations:
x – 2y = 10, 2x + y + 3z = 8 and -2y + z = 7
For matrix A = `[(2,5),(-11,7)]` (adj A)' is equal to:
For A = `[(3,1),(-1,2)]`, then 14A−1 is given by:
If `abs((2"x", -1),(4,2)) = abs ((3,0),(2,1))` then x is ____________.
