Advertisements
Advertisements
Question
Find the inverse of the matrices (if it exists).
`[(1,0,0),(3,3,0),(5,2,-1)]`
Advertisements
Solution
A = `[(1,0,0),(3,3,0),(5,2,-1)]`
|A| = `|(1,0,0),(3,3,0),(5,2,-1)|`
= −1[−3 − 0]
= 1 × (−3)
= −3
A11 = `(-1)^(1 + 1) |(3,0),(2,-1)|`
= (−1)2 [−3 − 0]
= 1 × (−3)
= −3
A12 = `(-1)^(1 + 2) |(3,0),(5,-1)|`
= (−1)3 [−3 − 0]
= −1 × (−3)
= 3
A13 = `(-1)^(1 + 3) |(3,3),(5,2)|`
= (−1)4 [6 − 15]
= 1 × (−9)
= −9
A21 = `(-1)^(2 + 1) |(0,0),(2,-1)|`
= (−1)3 [0 − 0]
= 0
A22 = `(-1)^(2 + 2) |(1,0),(5,-1)|`
= (−1)4 [−1 − 0]
= 1 × (−1)
= −1
A23 = `(-1)^(2 + 3) |(1,0),(5,2)|`
= (−1)5 [2 − 0]
= −1 × 2
= −2
A31 = `(-1)^(3 + 1) |(0,0),(3,0)|`
= (−1)4 [0 − 0]
= 0
A32 = `(-1)^(3 + 2) |(1,0),(3,3)|`
= (−1)5 [0 − 0]
= 0
A33 = `(-1)^(3 + 3) |(1,0),(3,3)|`
= (−1)6 [3 − 0]
= 1 × 3
= 3
∴ adj A = `[(-3,3,-9),(0,-1,-2),(0,0,3)] = [(-3,0,0),(3,-1,0),(-9,-2,3)]`
A−1 = `1/|A|` adj A
= `1/-3 [(-3,0,0),(3,-1,0),(-9,-2,3)]`
APPEARS IN
RELATED QUESTIONS
Find the inverse of the matrices (if it exists).
`[(2,-2),(4,3)]`
Find the inverse of the matrices (if it exists).
`[(1,0,0),(0, cos alpha, sin alpha),(0, sin alpha, -cos alpha)]`
If A is an invertible matrix of order 2, then det (A−1) is equal to ______.
Let A = `[(1, sin theta, 1),(-sin theta,1,sin theta),(-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.
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 \[A = \begin{bmatrix}4 & 3 \\ 2 & 5\end{bmatrix}\], find x and y such 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.
Find the matrix X satisfying the matrix equation \[X\begin{bmatrix}5 & 3 \\ - 1 & - 2\end{bmatrix} = \begin{bmatrix}14 & 7 \\ 7 & 7\end{bmatrix}\]
Find the matrix X for which
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}5 & 2 \\ 2 & 1\end{bmatrix}\]
Find the inverse by using elementary row transformations:
\[\begin{bmatrix}2 & 0 & - 1 \\ 5 & 1 & 0 \\ 0 & 1 & 3\end{bmatrix}\]
If adj \[A = \begin{bmatrix}2 & 3 \\ 4 & - 1\end{bmatrix}\text{ and adj }B = \begin{bmatrix}1 & - 2 \\ - 3 & 1\end{bmatrix}\]
If A is a square matrix, then write the matrix adj (AT) − (adj A)T.
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 \[A = \begin{bmatrix}1 & - 3 \\ 2 & 0\end{bmatrix}\], write adj A.
If A is an invertible matrix, then which of the following is not true ?
If \[A = \begin{bmatrix}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix}\] , then the value of |adj A| is _____________ .
If A5 = O such that \[A^n \neq I\text{ for }1 \leq n \leq 4,\text{ then }\left( I - A \right)^{- 1}\] equals ________ .
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 and B are invertible matrices, which of the following statement is not correct.
If \[A = \begin{bmatrix}1 & 0 & 1 \\ 0 & 0 & 1 \\ a & b & 2\end{bmatrix},\text{ then aI + bA + 2 }A^2\] 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 matrix A is such that \[3A^3 + 2 A^2 + 5 A + I = 0,\text{ then }A^{- 1}\] equal to _______________ .
Using matrix method, solve the following system of equations:
x – 2y = 10, 2x + y + 3z = 8 and -2y + z = 7
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)]`
Find the adjoint of the matrix A `= [(1,2),(3,4)].`
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 = [aij] is a square matrix of order 2 such that aij = `{(1"," "when i" ≠ "j"),(0"," "when" "i" = "j"):},` then A2 is ______.
If A = `[(2, -3, 5),(3, 2, -4),(1, 1, -2)]`, find A–1. Use A–1 to solve the following system of equations 2x − 3y + 5z = 11, 3x + 2y – 4z = –5, x + y – 2z = –3
If A = `[(0, 1),(0, 0)]`, then A2023 is equal to ______.
If for a square matrix A, A2 – A + I = 0, then A–1 equals ______.
