Advertisements
Advertisements
प्रश्न
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}1 & 2 & 0 \\ 2 & 3 & - 1 \\ 1 & - 1 & 3\end{bmatrix}\]
Advertisements
उत्तर
\[A = \begin{bmatrix}1 & 2 & 0 \\ 2 & 3 & - 1 \\ 1 & - 1 & 3\end{bmatrix}\]
We know
\[A = IA \]
\[ \Rightarrow \begin{bmatrix}1 & 2 & 0 \\ 2 & 3 & - 1 \\ 1 & - 1 & 3\end{bmatrix} = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} A\]
\[ \Rightarrow \begin{bmatrix}1 & 2 & 0 \\ 0 & - 1 & - 1 \\ 0 & - 3 & 3\end{bmatrix} = \begin{bmatrix}1 & 0 & 0 \\ - 2 & 1 & 0 \\ - 1 & 0 & 1\end{bmatrix} A \left[\text{ Applying }R_2 \to R_2 - 2 R_1\text{ and }R_3 \to R_3 - R_1 \right]\]
\[ \Rightarrow \begin{bmatrix}1 & 2 & 0 \\ 0 & 1 & 1 \\ 0 & - 3 & 3\end{bmatrix} = \begin{bmatrix}1 & 0 & 0 \\ 2 & - 1 & 0 \\ - 1 & 0 & 1\end{bmatrix} A \left[\text{ Applying }R_2 \to - R_2 \right]\]
\[ \Rightarrow \begin{bmatrix}1 & 0 & - 2 \\ 0 & 1 & 1 \\ 0 & 0 & 6\end{bmatrix} = \begin{bmatrix}- 3 & 2 & 0 \\ 2 & - 1 & 0 \\ 5 & - 3 & 1\end{bmatrix} A \left[\text{ Applying }R_1 \to R_1 - 2 R_2\text{ and }R_3 \to R_3 + 3 R_2 \right]\]
\[ \Rightarrow \begin{bmatrix}1 & 0 & - 2 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{bmatrix} = \begin{bmatrix}- 3 & 2 & 0 \\ 2 & - 1 & 0 \\ \frac{5}{6} & - \frac{1}{2} & \frac{1}{6}\end{bmatrix} A \left[\text{ Applying }R_3 \to \frac{1}{6} R_3 \right]\]
\[ \Rightarrow \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} = \begin{bmatrix}- \frac{4}{3} & 1 & \frac{1}{3} \\ \frac{7}{6} & - \frac{1}{2} & \frac{- 1}{6} \\ \frac{5}{6} & - \frac{1}{2} & \frac{1}{6}\end{bmatrix}A \left[\text{ Applying }R_1 \to R_1 + 2 R_3\text{ and }R_2 \to R_2 - R_3 \right]\]
\[ \therefore A^{- 1} = \begin{bmatrix}- \frac{4}{3} & 1 & \frac{1}{3} \\ \frac{7}{6} & - \frac{1}{2} & \frac{- 1}{6} \\ \frac{5}{6} & - \frac{1}{2} & \frac{1}{6}\end{bmatrix}\]
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)]`
Verify A(adj A) = (adj A)A = |A|I.
`[(2,3),(-4,-6)]`
Find the inverse of the matrices (if it exists).
`[(-1,5),(-3,2)]`
For the matrix A = `[(3,2),(1,1)]` find the numbers a and b such that A2 + aA + bI = 0.
If A = `[(2,-1,1),(-1,2,-1),(1,-1,2)]` verify that A3 − 6A2 + 9A − 4I = 0 and hence find A−1.
If A is an invertible matrix of order 2, then det (A−1) is equal to ______.
If \[A = \begin{bmatrix}- 4 & - 3 & - 3 \\ 1 & 0 & 1 \\ 4 & 4 & 3\end{bmatrix}\], show that adj A = A.
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 \[\left( AB \right)^{- 1} = B^{- 1} A^{- 1} :\]
\[A = \begin{bmatrix}2 & 1 \\ 5 & 3\end{bmatrix}\text{ and }B \begin{bmatrix}4 & 5 \\ 3 & 4\end{bmatrix}\]
Let \[A = \begin{bmatrix}3 & 2 \\ 7 & 5\end{bmatrix}\text{ and }B = \begin{bmatrix}6 & 7 \\ 8 & 9\end{bmatrix} .\text{ Find }\left( AB \right)^{- 1}\]
Let
\[F \left( \alpha \right) = \begin{bmatrix}\cos \alpha & - \sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{bmatrix}\text{ and }G\left( \beta \right) = \begin{bmatrix}\cos \beta & 0 & \sin \beta \\ 0 & 1 & 0 \\ - \sin \beta & 0 & \cos \beta\end{bmatrix}\]
Show that
If \[A = \begin{bmatrix}2 & 3 \\ 1 & 2\end{bmatrix}\] , verify that \[A^2 - 4 A + I = O,\text{ where }I = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\text{ and }O = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\] . Hence, find A−1.
If \[A = \begin{bmatrix}3 & 1 \\ - 1 & 2\end{bmatrix}\], show that
Show that \[A = \begin{bmatrix}5 & 3 \\ - 1 & - 2\end{bmatrix}\] satisfies the equation \[x^2 - 3x - 7 = 0\]. Thus, find A−1.
prove that \[A^{- 1} = A^3\]
Find the matrix X for which
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}3 & 10 \\ 2 & 7\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}2 & - 1 & 4 \\ 4 & 0 & 7 \\ 3 & - 2 & 7\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}- 1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix}\]
If A is a non-singular symmetric matrix, write whether A−1 is symmetric or skew-symmetric.
If A is a singular matrix, then adj A is ______.
If A, B are two n × n non-singular matrices, then __________ .
If \[A = \begin{bmatrix}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix}\] , then the value of |adj A| is _____________ .
If d is the determinant of a square matrix A of order n, then the determinant of its adjoint is _____________ .
If \[A^2 - A + I = 0\], then the inverse of A is __________ .
If \[A = \begin{bmatrix}2 & 3 \\ 5 & - 2\end{bmatrix}\] be such that \[A^{- 1} = kA\], then k equals ___________ .
Find A−1, if \[A = \begin{bmatrix}1 & 2 & 5 \\ 1 & - 1 & - 1 \\ 2 & 3 & - 1\end{bmatrix}\] . Hence solve the following system of linear equations:x + 2y + 5z = 10, x − y − z = −2, 2x + 3y − z = −11
`("aA")^-1 = 1/"a" "A"^-1`, where a is any real number and A is a square matrix.
|A–1| ≠ |A|–1, where A is non-singular matrix.
A square matrix A is invertible if det A is equal to ____________.
Find the adjoint of the matrix A `= [(1,2),(3,4)].`
If A = `[(0, 1),(0, 0)]`, then A2023 is equal to ______.
| To raise money for an orphanage, students of three schools A, B and C organised an exhibition in their residential colony, where they sold paper bags, scrap books and pastel sheets made by using recycled paper. Student of school A sold 30 paper bags, 20 scrap books and 10 pastel sheets and raised ₹ 410. Student of school B sold 20 paper bags, 10 scrap books and 20 pastel sheets and raised ₹ 290. Student of school C sold 20 paper bags, 20 scrap books and 20 pastel sheets and raised ₹ 440. |
Answer the following question:
- Translate the problem into a system of equations.
- Solve the system of equation by using matrix method.
- Hence, find the cost of one paper bag, one scrap book and one pastel sheet.
