Advertisements
Advertisements
प्रश्न
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}- 1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix}\]
Advertisements
उत्तर
\[\text{ Let }A = \begin{bmatrix}- 1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix} . \]
To find inverse, first write A = IA .
\[i . e . , \begin{bmatrix}- 1 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix} = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}A\]
\[ \Rightarrow \begin{bmatrix}1 & - 1 & - 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix} = \begin{bmatrix}- 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}A \left[\text{ Applying }R_1 \to \left( - 1 \right) R_1 \right]\]
\[ \Rightarrow \begin{bmatrix}1 & - 1 & - 2 \\ 0 & 3 & 5 \\ 0 & 4 & 7\end{bmatrix} = \begin{bmatrix}- 1 & 0 & 0 \\ 1 & 1 & 0 \\ 3 & 0 & 1\end{bmatrix}A \left[\text{ Applying }R_2 \to R_2 - R_1\text{ and }R_3 \to R_3 - 3 R_1 \right]\]
\[ \Rightarrow \begin{bmatrix}1 & - 1 & - 2 \\ 0 & 1 & \frac{5}{3} \\ 0 & 4 & 7\end{bmatrix} = \begin{bmatrix}- 1 & 0 & 0 \\ \frac{1}{3} & \frac{1}{3} & 0 \\ 3 & 0 & 1\end{bmatrix}A \left[\text{ Applying }R_2 \to \frac{1}{3} R_2 \right]\]
\[ \Rightarrow \begin{bmatrix}1 & 0 & \frac{- 1}{3} \\ 0 & 1 & \frac{5}{3} \\ 0 & 0 & \frac{1}{3}\end{bmatrix} = \begin{bmatrix}\frac{- 2}{3} & \frac{1}{3} & 0 \\ \frac{1}{3} & \frac{1}{3} & 0 \\ \frac{5}{3} & - \frac{4}{3} & 1\end{bmatrix}A \left[\text{ Applying }R_3 \to R_3 - 4 R_2\text{ and }R_1 \to R_1 + R_2 \right]\]
\[ \Rightarrow \begin{bmatrix}1 & 0 & \frac{- 1}{3} \\ 0 & 1 & \frac{5}{3} \\ 0 & 0 & 1\end{bmatrix} = \begin{bmatrix}\frac{- 2}{3} & \frac{1}{3} & 0 \\ \frac{1}{3} & \frac{1}{3} & 0 \\ 5 & - 4 & 3\end{bmatrix}A \left[\text{ Applying }R_3 \to 3 R_3 \right]\]
\[ \Rightarrow \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} = \begin{bmatrix}1 & - 1 & 1 \\ - 8 & 7 & - 5 \\ 5 & - 4 & 3\end{bmatrix}A \left[\text{ Applying }R_2 \to R_2 - \frac{5}{3} R_3\text{ and }R_1 \to R_1 + \frac{1}{3} R_3 \right]\]
\[\text{ Hence, }A^{- 1} = \begin{bmatrix}1 & - 1 & 1 \\ - 8 & 7 & - 5 \\ 5 & - 4 & 3\end{bmatrix} .\]
APPEARS IN
संबंधित 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?
Find the inverse of the matrices (if it exists).
`[(1,-1,2),(0,2,-3),(3,-2,4)]`
If x, y, z are nonzero real numbers, then the inverse of matrix A = `[(x,0,0),(0,y,0),(0,0,z)]` is ______.
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.
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
For the matrix \[A = \begin{bmatrix}1 & 1 & 1 \\ 1 & 2 & - 3 \\ 2 & - 1 & 3\end{bmatrix}\] . Show that
Find the matrix X satisfying the equation
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:
\[\begin{bmatrix}7 & 1 \\ 4 & - 3\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}5 & 2 \\ 2 & 1\end{bmatrix}\]
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:
\[\begin{bmatrix}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}3 & - 3 & 4 \\ 2 & - 3 & 4 \\ 0 & - 1 & 1\end{bmatrix}\]
If A is symmetric matrix, write whether AT is symmetric or skew-symmetric.
If A is a non-singular symmetric matrix, write whether A−1 is symmetric or skew-symmetric.
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 \[\begin{bmatrix}3 & - 2 \\ - 7 & 5\end{bmatrix} .\]
If \[A = \begin{bmatrix}3 & 4 \\ 2 & 4\end{bmatrix}, B = \begin{bmatrix}- 2 & - 2 \\ 0 & - 1\end{bmatrix},\text{ then }\left( A + B \right)^{- 1} =\]
For any 2 × 2 matrix, if \[A \left( adj A \right) = \begin{bmatrix}10 & 0 \\ 0 & 10\end{bmatrix}\] , then |A| is equal to ______ .
If A satisfies the equation \[x^3 - 5 x^2 + 4x + \lambda = 0\] then A-1 exists if _____________ .
If \[A = \begin{bmatrix}2 & 3 \\ 5 & - 2\end{bmatrix}\] be such that \[A^{- 1} = kA\], then k equals ___________ .
If \[\begin{bmatrix}1 & - \tan \theta \\ \tan \theta & 1\end{bmatrix} \begin{bmatrix}1 & \tan \theta \\ - \tan \theta & 1\end{bmatrix} - 1 = \begin{bmatrix}a & - b \\ b & a\end{bmatrix}\], then _______________ .
If A = `[(x, 5, 2),(2, y, 3),(1, 1, z)]`, xyz = 80, 3x + 2y + 10z = 20, ten A adj. A = `[(81, 0, 0),(0, 81, 0),(0, 0, 81)]`
|A–1| ≠ |A|–1, where A is non-singular matrix.
If A, B be two square matrices such that |AB| = O, then ____________.
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 = `[(0, 1),(0, 0)]`, then A2023 is equal to ______.
Given that A is a square matrix of order 3 and |A| = –2, then |adj(2A)| is equal to ______.
