Advertisements
Advertisements
Question
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}2 & - 1 & 4 \\ 4 & 0 & 7 \\ 3 & - 2 & 7\end{bmatrix}\]
Solution
\[A = \begin{bmatrix}2 & - 1 & 4 \\ 4 & 0 & 2 \\ 3 & - 2 & 7\end{bmatrix}\]
We know
\[A = IA \]
\[ \Rightarrow \begin{bmatrix}2 & - 1 & 4 \\ 4 & 0 & 2 \\ 3 & - 2 & 7\end{bmatrix} = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} A\]
\[ \Rightarrow \begin{bmatrix}1 & - \frac{1}{2} & 2 \\ 4 & 0 & 2 \\ 3 & - 2 & 7\end{bmatrix} = \begin{bmatrix}\frac{1}{2} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} A \left[\text{ Applying } R_1 \to \frac{1}{2} R_1 \right]\]
\[ \Rightarrow \begin{bmatrix}1 & - \frac{1}{2} & 2 \\ 0 & 2 & - 6 \\ 0 & - \frac{1}{2} & 1\end{bmatrix} = \begin{bmatrix}\frac{1}{2} & 0 & 0 \\ - 2 & 1 & 0 \\ \frac{- 3}{2} & 0 & 1\end{bmatrix}A \left[\text{ Applying }R_2 \to R_2 - 4 R_1\text{ and }R_3 \to R_3 - 3 R_1 \right]\]
\[ \Rightarrow \begin{bmatrix}1 & - \frac{1}{2} & 2 \\ 0 & 1 & - 3 \\ 0 & - \frac{1}{2} & 1\end{bmatrix} = \begin{bmatrix}\frac{1}{2} & 0 & 0 \\ - 1 & \frac{1}{2} & 0 \\ \frac{- 3}{2} & 0 & 1\end{bmatrix} A \left[\text{ Applying }R_2 \to \frac{1}{2} R_2 \right]\]
\[ \Rightarrow \begin{bmatrix}1 & 0 & \frac{1}{2} \\ 0 & 1 & - 3 \\ 0 & 0 & - \frac{1}{2}\end{bmatrix} = \begin{bmatrix}0 & \frac{1}{4} & 0 \\ - 1 & \frac{1}{2} & 0 \\ - 2 & \frac{1}{4} & 1\end{bmatrix} A \left[\text{ Applying }R_1 \to R_1 + \frac{1}{2} R_2\text{ and }R_3 \to R_3 + \frac{1}{2} R_2 \right]\]
\[ \Rightarrow \begin{bmatrix}1 & 0 & \frac{1}{2} \\ 0 & 1 & - 3 \\ 0 & 0 & 1\end{bmatrix} = \begin{bmatrix}0 & \frac{1}{4} & 0 \\ - 1 & \frac{1}{2} & 0 \\ 4 & - \frac{1}{2} & - 2\end{bmatrix} A \left[\text{ Applying }R_3 \to - 2 R_3 \right]\]
\[ \Rightarrow \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} = \begin{bmatrix}- 2 & \frac{1}{2} & 1 \\ 11 & - 1 & - 6 \\ 4 & - \frac{1}{2} & - 2\end{bmatrix} A \left[\text{ Applying }R_1 \to R_1 - \frac{1}{2} R_3\text{ and }R_2 \to R_2 + 3 R_3 \right]\]
\[ \Rightarrow A^{- 1} = \begin{bmatrix}- 2 & \frac{1}{2} & 1 \\ 11 & - 1 & - 6 \\ 4 & - \frac{1}{2} & - 2\end{bmatrix} \]
APPEARS IN
RELATED QUESTIONS
Find the adjoint of the matrices.
`[(1,2),(3,4)]`
Let `A =[(3,7),(2,5)] and B = [(6,8),(7,9)]`. Verify that `(AB)^(-1) = B^(-1)A^(-1).`
For the matrix A = `[(1,1,1),(1,2,-3),(2,-1,3)]` show that A3 − 6A2 + 5A + 11 I = O. Hence, find A−1.
If A = `[(2,-1,1),(-1,2,-1),(1,-1,2)]` verify that A3 − 6A2 + 9A − 4I = O and 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)`
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:
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.
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.
Find the inverse of the following matrix.
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.
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
Show that
If \[A = \begin{bmatrix}4 & 3 \\ 2 & 5\end{bmatrix}\], find x and y such that
Show that \[A = \begin{bmatrix}6 & 5 \\ 7 & 6\end{bmatrix}\] satisfies the equation \[x^2 - 12x + 1 = O\]. Thus, find A−1.
Verify that \[A^3 - 6 A^2 + 9A - 4I = O\] and hence find A−1.
If \[A = \begin{bmatrix}3 & - 3 & 4 \\ 2 & - 3 & 4 \\ 0 & - 1 & 1\end{bmatrix}\] , show that \[A^{- 1} = A^3\]
Find the matrix X for which
Find the adjoint of the matrix \[A = \begin{bmatrix}- 1 & - 2 & - 2 \\ 2 & 1 & - 2 \\ 2 & - 2 & 1\end{bmatrix}\] and hence show that \[A\left( adj A \right) = \left| A \right| I_3\].
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}5 & 2 \\ 2 & 1\end{bmatrix}\]
If A is a non-singular symmetric matrix, write whether A−1 is symmetric or skew-symmetric.
If \[A = \begin{bmatrix}\cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{bmatrix}\text{ and }A \left( adj A = \right)\begin{bmatrix}k & 0 \\ 0 & k\end{bmatrix}\], then find the value of k.
If \[A = \begin{bmatrix}a & b \\ c & d\end{bmatrix}, B = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\] , find adj (AB).
If \[A = \begin{bmatrix}2 & 3 \\ 5 & - 2\end{bmatrix}\] , write \[A^{- 1}\] in terms of A.
If \[S = \begin{bmatrix}a & b \\ c & d\end{bmatrix}\], then adj A is ____________ .
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 A5 = O such that \[A^n \neq I\text{ for }1 \leq n \leq 4,\text{ then }\left( I - A \right)^{- 1}\] equals ________ .
(a) 3
(b) 0
(c) − 3
(d) 1
`("aA")^-1 = 1/"a" "A"^-1`, where a is any real number and A is a square matrix.
The value of `abs (("cos" (alpha + beta),-"sin" (alpha + beta),"cos" 2 beta),("sin" alpha, "cos" alpha, "sin" beta),(-"cos" alpha, "sin" alpha, "cos" beta))` is independent of ____________.
For matrix A = `[(2,5),(-11,7)]` (adj A)' is equal to:
If `abs((2"x", -1),(4,2)) = abs ((3,0),(2,1))` then x is ____________.
If A is a square matrix of order 3, |A′| = −3, then |AA′| = ______.
