Advertisements
Advertisements
Question
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}1 & 1 & 2 \\ 3 & 1 & 1 \\ 2 & 3 & 1\end{bmatrix}\]
Solution
\[A = \begin{bmatrix}1 & 1 & 2 \\ 3 & 1 & 1 \\ 2 & 3 & 1\end{bmatrix}\]
We know
\[A = IA \]
\[ \Rightarrow \begin{bmatrix}1 & 1 & 2 \\ 3 & 1 & 1 \\ 2 & 3 & 1\end{bmatrix} = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} A\]
\[ \Rightarrow \begin{bmatrix}1 & 1 & 2 \\ 0 & - 2 & - 5 \\ 0 & 1 & - 3\end{bmatrix} = \begin{bmatrix}1 & 0 & 0 \\ - 3 & 1 & 0 \\ - 2 & 0 & 1\end{bmatrix}A \left[\text{ Applying }R_2 \to R_2 - 3 R_1\text{ and }R_3 \to R_3 - 2 R_1 \right]\]
\[ \Rightarrow \begin{bmatrix}1 & 1 & 2 \\ 0 & 1 & \frac{5}{2} \\ 0 & 1 & - 3\end{bmatrix} = \begin{bmatrix}1 & 0 & 0 \\ \frac{3}{2} & - \frac{1}{2} & 0 \\ - 2 & 0 & 1\end{bmatrix} A \left[\text{ Applying }R_2 \to \frac{- 1}{2} R_2 \right]\]
\[ \Rightarrow \begin{bmatrix}1 & 0 & - \frac{1}{2} \\ 0 & 1 & \frac{5}{2} \\ 0 & 0 & - \frac{11}{2}\end{bmatrix} = \begin{bmatrix}- \frac{1}{2} & \frac{1}{2} & 0 \\ \frac{3}{2} & - \frac{1}{2} & 0 \\ - \frac{7}{2} & \frac{1}{2} & 1\end{bmatrix} A \left[\text{ Applying }R_1 \to R_1 - R_2\text{ and }R_3 \to R_3 - R_2 \right]\]
\[ \Rightarrow \begin{bmatrix}1 & 0 & - \frac{1}{2} \\ 0 & 1 & \frac{5}{2} \\ 0 & 0 & 1\end{bmatrix} = \begin{bmatrix}- \frac{1}{2} & \frac{1}{2} & 0 \\ \frac{3}{2} & - \frac{1}{2} & 0 \\ \frac{7}{11} & \frac{- 1}{11} & \frac{- 2}{11}\end{bmatrix} A \left[\text{ Applying }R_3 \to - \frac{2}{11} R_3 \right]\]
\[ \Rightarrow \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} = \begin{bmatrix}\frac{- 2}{11} & \frac{5}{11} & \frac{- 1}{11} \\ \frac{- 1}{11} & \frac{- 3}{11} & \frac{5}{11} \\ \frac{7}{11} & \frac{- 1}{11} & \frac{- 2}{11}\end{bmatrix} A \left[\text{ Applying }R_2 \to R_2 - \frac{5}{2} R_3\text{ and }R_1 \to R_1 + \frac{1}{2} R_3 \right]\]
\[ \Rightarrow A^{- 1} = \frac{1}{11}\begin{bmatrix}- 2 & 5 & - 1 \\ - 1 & - 3 & 5 \\ 7 & - 1 & - 2\end{bmatrix} \]
APPEARS IN
RELATED QUESTIONS
Find the adjoint of the matrices.
`[(1,2),(3,4)]`
Find the adjoint of the matrices.
`[(1,-1,2),(2,3,5),(-2,0,1)]`
If A = `[(3,1),(-1,2)]` show that A2 – 5A + 7I = 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
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 1),(-1, -sin theta, 1)]` where 0 ≤ θ≤ 2π, then ______.
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.
Find the inverse of the following matrix and verify that \[A^{- 1} A = I_3\]
Let \[A = \begin{bmatrix}3 & 2 \\ 7 & 5\end{bmatrix}\text{ and }B = \begin{bmatrix}6 & 7 \\ 8 & 9\end{bmatrix} .\text{ Find }\left( AB \right)^{- 1}\]
Find the inverse of the matrix \[A = \begin{bmatrix}a & b \\ c & \frac{1 + bc}{a}\end{bmatrix}\] and show that \[a A^{- 1} = \left( a^2 + bc + 1 \right) I - aA .\]
Show that \[A = \begin{bmatrix}5 & 3 \\ - 1 & - 2\end{bmatrix}\] satisfies the equation \[x^2 - 3x - 7 = 0\]. Thus, find A−1.
For the matrix \[A = \begin{bmatrix}1 & 1 & 1 \\ 1 & 2 & - 3 \\ 2 & - 1 & 3\end{bmatrix}\] . Show that
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:
\[\begin{bmatrix}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}1 & 2 & 0 \\ 2 & 3 & - 1 \\ 1 & - 1 & 3\end{bmatrix}\]
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:
\[\begin{bmatrix}3 & 0 & - 1 \\ 2 & 3 & 0 \\ 0 & 4 & 1\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}1 & 3 & - 2 \\ - 3 & 0 & - 1 \\ 2 & 1 & 0\end{bmatrix}\]
If A is symmetric matrix, write whether AT 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.
If \[A = \begin{bmatrix}1 & - 3 \\ 2 & 0\end{bmatrix}\], write adj A.
If \[A = \begin{bmatrix}3 & 1 \\ 2 & - 3\end{bmatrix}\], then find |adj A|.
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 non-singular square matrix A, B and C of the same order \[\left( A B^{- 1} C \right) =\] ______________ .
(a) 3
(b) 0
(c) − 3
(d) 1
If \[A = \begin{bmatrix}2 & - 3 & 5 \\ 3 & 2 & - 4 \\ 1 & 1 & - 2\end{bmatrix}\], find A−1 and hence solve the system of linear equations 2x − 3y + 5z = 11, 3x + 2y − 4z = −5, x + y + 2z = −3
(A3)–1 = (A–1)3, where A is a square matrix and |A| ≠ 0.
A square matrix A is invertible if det A is equal to ____________.
If A = [aij] is a square matrix of order 2 such that aij = `{(1"," "when i" ≠ "j"),(0"," "when" "i" = "j"):},` then A2 is ______.
