Advertisements
Advertisements
प्रश्न
The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves Rs 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value?
Advertisements
उत्तर
Let the monthly incomes of Aryan and Babban be 3x and 4x, respectively.
Suppose their monthly expenditures are 5y and 7y, respectively.
Since each saves Rs 15,000 per month
Monthly saving of Aryan: 3x−5y=15,000
Monthly saving of Babban: 4x−7y=15,000
The above system of equations can be written in the matrix form as follows:
`[(3,5),(4,-7)][(x),(y)]=[(15000),(15000)]`
or
AX = B, where
`A=[(3,-5),(4,-7)],X=[(x),(y)]`
Now,
`|A|=|(3,-5),(4,-7)|=-21-(-20)=-1`
Adj `A=[(-7,-4),(5,3)]^T=[(-7,5),(-4,3)]`
So,
`A^(-1)=1/|A|adjA=-1[(-7,5),(-4,3)]=[(7,-5),(4,-3)]`
∴ X = A-1B
`=>[(x),(y)]=[(7,-5),(4,-3)][(15000),(15000)]`
`=>[(x),(y)]=[(105000,-75000),(60000,-45000)]`
`=>[(x),(y)]=[(30000),(15000)]`
⇒ x=30,000 and y=15,000
Therefore,
Monthly income of Aryan = 3×Rs 30,000=Rs 90,000
Monthly income of Babban = 4×Rs 30,000= Rs 1,20,000
From this problem, we are encouraged to understand the power of savings. We should save certain part of our monthly income for the future
APPEARS IN
संबंधित प्रश्न
Verify A(adj A) = (adj A)A = |A|I.
`[(2,3),(-4,-6)]`
Find the inverse of the matrices (if it exists).
`[(2,-2),(4,3)]`
For the matrix A = `[(3,2),(1,1)]` find the numbers a and b such that A2 + aA + bI = 0.
Find the adjoint of the following matrix:
\[\begin{bmatrix}\cos \alpha & \sin \alpha \\ \sin \alpha & \cos \alpha\end{bmatrix}\]
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:
\[\begin{bmatrix}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{bmatrix}\]
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.
For the matrix
Find the inverse of the following matrix.
Find the inverse of the following matrix and verify that \[A^{- 1} A = I_3\]
For the following pair of matrix verify that \[\left( AB \right)^{- 1} = B^{- 1} A^{- 1} :\]
\[A = \begin{bmatrix}3 & 2 \\ 7 & 5\end{bmatrix}\text{ and }B \begin{bmatrix}4 & 6 \\ 3 & 2\end{bmatrix}\]
Given \[A = \begin{bmatrix}5 & 0 & 4 \\ 2 & 3 & 2 \\ 1 & 2 & 1\end{bmatrix}, B^{- 1} = \begin{bmatrix}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{bmatrix}\] . Compute (AB)−1.
Show that
prove that \[A^{- 1} = A^3\]
Solve the matrix equation \[\begin{bmatrix}5 & 4 \\ 1 & 1\end{bmatrix}X = \begin{bmatrix}1 & - 2 \\ 1 & 3\end{bmatrix}\], where X is a 2 × 2 matrix.
If \[A = \begin{bmatrix}1 & - 2 & 3 \\ 0 & - 1 & 4 \\ - 2 & 2 & 1\end{bmatrix},\text{ find }\left( A^T \right)^{- 1} .\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}2 & 5 \\ 1 & 3\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}3 & 10 \\ 2 & 7\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}2 & - 1 & 4 \\ 4 & 0 & 7 \\ 3 & - 2 & 7\end{bmatrix}\]
If \[A = \begin{bmatrix}2 & 3 \\ 5 & - 2\end{bmatrix}\] be such that \[A^{- 1} = k A,\] then find the value of k.
Find the inverse of the matrix \[\begin{bmatrix} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{bmatrix}\]
If B is a non-singular matrix and A is a square matrix, then det (B−1 AB) is equal to ___________ .
If A = `[(0, 1, 3),(1, 2, x),(2, 3, 1)]`, A–1 = `[(1/2, -4, 5/2),(-1/2, 3, -3/2),(1/2, y, 1/2)]` then x = 1, y = –1.
If A and B are invertible matrices, then which of the following is not correct?
Find the value of x for which the matrix A `= [(3 - "x", 2, 2),(2,4 - "x", 1),(-2,- 4,-1 - "x")]` is singular.
If the equation a(y + z) = x, b(z + x) = y, c(x + y) = z have non-trivial solutions then the value of `1/(1+"a") + 1/(1+"b") + 1/(1+"c")` is ____________.
If A = `[(2, -3, 5),(3, 2, -4),(1, 1, -2)]`, find A–1. Use A–1 to solve the following system of equations 2x − 3y + 5z = 11, 3x + 2y – 4z = –5, x + y – 2z = –3
If A = `[(0, 1),(0, 0)]`, then A2023 is equal to ______.
