Advertisements
Advertisements
Question
Show that the matrix, \[A = \begin{bmatrix}1 & 0 & - 2 \\ - 2 & - 1 & 2 \\ 3 & 4 & 1\end{bmatrix}\] satisfies the equation, \[A^3 - A^2 - 3A - I_3 = O\] . Hence, find A−1.
Advertisements
Solution
\[\text{ We have, }A = \begin{bmatrix} 1 & 0 &- 2\\ - 2 & - 1 & 2\\ 3 & 4 & 1 \end{bmatrix} \]
\[ \Rightarrow \left| A \right| = \begin{vmatrix}| 1 & 0 &- 2\\ - 2 & - 1 & 2\\ 3 & 4 & 1 \end{vmatrix} = 1\left( - 9 \right) + 0 - 2\left( - 8 \right) = - 9 + 16 = 7 \]
\[\text{ Since, }\left| A \right| \neq 0\]
\[\text{ Hence, }A^{- 1}\text{ exists .} \]
Now,
\[ A^2 = \begin{bmatrix} 1 & 0 & - 2\\ - 2 & - 1 & 2\\ 3 & 4 & 1 \end{bmatrix}\begin{bmatrix} 1 & 0 &- 2\\ - 2 & - 1 & 2\\ 3 & 4 & 1 \end{bmatrix} = \begin{bmatrix} 1 + 0 - 6 & 0 + 0 - 8 & - 2 + 0 - 2\\ - 2 + 2 + 6 & 0 + 1 + 8 & 4 - 2 + 2\\ 3 - 8 + 3 & 0 - 4 + 4 & - 6 + 8 + 1 \end{bmatrix} = \begin{bmatrix} - 5 & - 8 & - 4\\ 6 & 9 & 4\\ - 2 & 0 & 3 \end{bmatrix}\]
\[ A^3 = A^2 . A = \begin{bmatrix} - 5 & - 8 & - 4\\ 6 & 9 & 4\\ - 2 & 0 & 3 \end{bmatrix}\begin{bmatrix} 1 & 0 & - 2\\ - 2 & - 1 & 2\\ 3 & 4 & 1 \end{bmatrix} = \begin{bmatrix} - 5 + 16 - 12 & 0 + 8 - 16 & 10 - 16 - 4\\ 6 - 18 + 12 & 0 - 9 + 16 & - 12 + 18 + 4\\ - 2 + 0 + 9 & 0 + 0 + 12 & 4 + 0 + 3 \end{bmatrix} = \begin{bmatrix} - 1 & - 8 & - 10\\ 0 & 7 & 10\\ 7 & 12 & 7 \end{bmatrix} \]
\[\text{ Now, }A^3 - A^2 - 3A - I_3 = \begin{bmatrix} - 1 & - 8 & - 10\\ 0 & 7 & 10\\ 7 & 12 & 7 \end{bmatrix} - \begin{bmatrix} - 5 & - 8 & - 4\\ 6 & 9 & 4\\ - 2 & 0 & 3 \end{bmatrix} - 3\begin{bmatrix} 1 & 0 & - 2\\ - 2 & - 1 & 2\\ 3 & 4 & 1 \end{bmatrix} - \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} \\ = \begin{bmatrix} - 1 + 5 - 3 - 1 & - 8 + 8 + 0 + 0 & - 10 + 4 + 6 - 0\\0 - 6 + 6 - 0 & 7 - 9 + 3 - 1 & 10 - 4 - 6 - 0\\ 7 + 2 - 9 - 0 & 12 + 0 - 12 - 0 & 7 - 3 - 3 - 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix} = O\]
Hence proved .
\[\text{ Now,} A^3 - A^2 - 3A - I_3 = O (\text{ Null matrix })\]
\[ \Rightarrow A^{- 1} \left( A^3 - A^2 - 3A - I_3 \right) = A^{- 1} O (\text{ Pre - multiplying by A}^{- 1} )\]
\[ \Rightarrow A^2 - A^1 - 3 I_3 = A^{- 1} \]
\[ \Rightarrow \begin{bmatrix} - 5 & - 8 & - 4\\ 6 & 9 & 4\\ - 2 & 0 & 3 \end{bmatrix} - \begin{bmatrix} 1 & 0 & - 2\\ - 2 & - 1 & 2\\ 3 & 4 & 1 \end{bmatrix} - 3\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} = A^{- 1} \]
\[ \Rightarrow \begin{bmatrix} - 5 - 1 - 3 & - 8 - 0 - 0 & - 4 + 2 + 0\\ 6 + 2 + 0 & 9 + 1 - 3 & 4 - 2\\- 2 - 3 - 0 & 0 - 4 - 0 & 3 - 1 - 3 \end{bmatrix} = \begin{bmatrix} - 9 & - 8 & - 2\\ 8 & 7 &2\\ - 5 & - 4 & - 1 \end{bmatrix} = A^{- 1} \]
\[ \Rightarrow A^{- 1} = \begin{bmatrix} - 9 & - 8 & - 2\\ 8 & 7 & 2\\ - 5 & - 4 & - 1 \end{bmatrix}\]
APPEARS IN
RELATED QUESTIONS
Find the inverse of the matrices (if it exists).
`[(-1,5),(-3,2)]`
Find the inverse of the matrices (if it exists).
`[(2,1,3),(4,-1,0),(-7,2,1)]`
Find the adjoint of the following matrix:
\[\begin{bmatrix}- 3 & 5 \\ 2 & 4\end{bmatrix}\]
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.
Find the inverse of the following matrix:
Find the inverse of the following matrix.
Find the inverse of the following matrix.
For the following pair of matrix verify that \[\left( AB \right)^{- 1} = B^{- 1} A^{- 1} :\]
\[A = \begin{bmatrix}3 & 2 \\ 7 & 5\end{bmatrix}\text{ and }B \begin{bmatrix}4 & 6 \\ 3 & 2\end{bmatrix}\]
For the following pair of matrix verify that \[\left( AB \right)^{- 1} = B^{- 1} A^{- 1} :\]
\[A = \begin{bmatrix}2 & 1 \\ 5 & 3\end{bmatrix}\text{ and }B \begin{bmatrix}4 & 5 \\ 3 & 4\end{bmatrix}\]
Given \[A = \begin{bmatrix}2 & - 3 \\ - 4 & 7\end{bmatrix}\], compute A−1 and show that \[2 A^{- 1} = 9I - A .\]
Show that \[A = \begin{bmatrix}5 & 3 \\ - 1 & - 2\end{bmatrix}\] satisfies the equation \[x^2 - 3x - 7 = 0\]. Thus, find A−1.
Show that \[A = \begin{bmatrix}6 & 5 \\ 7 & 6\end{bmatrix}\] satisfies the equation \[x^2 - 12x + 1 = O\]. Thus, find A−1.
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 \[A^2 - 4A - 5I = O\]
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:
\[\begin{bmatrix}3 & - 3 & 4 \\ 2 & - 3 & 4 \\ 0 & - 1 & 1\end{bmatrix}\]
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:
\[\begin{bmatrix}3 & 0 & - 1 \\ 2 & 3 & 0 \\ 0 & 4 & 1\end{bmatrix}\]
If A is symmetric matrix, write whether AT is symmetric or skew-symmetric.
If A is a square matrix, then write the matrix adj (AT) − (adj A)T.
If A is an invertible matrix such that |A−1| = 2, find the value of |A|.
If for the matrix A, A3 = I, then A−1 = _____________ .
For non-singular square matrix A, B and C of the same order \[\left( A B^{- 1} C \right) =\] ______________ .
If \[A^2 - A + I = 0\], then the inverse of A is __________ .
If A and B are invertible matrices, which of the following statement is not correct.
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
If A, B be two square matrices such that |AB| = O, then ____________.
Find the adjoint of the matrix A `= [(1,2),(3,4)].`
If A = [aij] is a square matrix of order 2 such that aij = `{(1"," "when i" ≠ "j"),(0"," "when" "i" = "j"):},` then A2 is ______.
If `abs((2"x", -1),(4,2)) = abs ((3,0),(2,1))` then x is ____________.
Read the following passage:
|
Gautam buys 5 pens, 3 bags and 1 instrument box and pays a sum of ₹160. From the same shop, Vikram buys 2 pens, 1 bag and 3 instrument boxes and pays a sum of ₹190. Also, Ankur buys 1 pen, 2 bags and 4 instrument boxes and pays a sum of ₹250. |
Based on the above information, answer the following questions:
- Convert the given above situation into a matrix equation of the form AX = B. (1)
- Find | A |. (1)
- Find A–1. (2)
OR
Determine P = A2 – 5A. (2)
