English

Find the adjoint of the matrices. [(1,-1,2),(2,3,5),(-2,0,1)] - Mathematics

Advertisements
Advertisements

Question

Find the adjoint of the matrices.

`[(1,-1,2),(2,3,5),(-2,0,1)]`

Sum
Advertisements

Solution

Let A = `[(1,-1,2),(2,3,5),(-2,0,1)]`

Let Cij be cofactors of aij in A.

C11 = `(-1)^(1 + 1) |(3,5),(0,1)|`

= 3 − 0

= 3

C12 = `(-1)^(1 + 2) |(2,5),(-2,1)|`

= −(2 + 10)

= −12

C13 = `(-1)^(1 + 3) |(2,3),(-2,0)|`

= 0 + 6

= 6

C21 = `(-1)^(2 + 1) |(-1,2),(0,1)|`

= −(−1 − 0)

= 1

C22 = `(-1)^(2 + 2) |(1,2),(-2,1)|`

= 1 + 4

= 5

C23 = `(-1)^(2 + 3) |(1,-1),(-2,0)|`

= −(0 − 2)

= 2

C31 = `(-1)^(3 + 1) |(-1,2),(3,5)|`

= −5 − 6

= −11

C32 = `(-1)^(3 + 2) |(1,2),(2,5)|`

= −(5 − 4)

= −l

C33 = `(-1)^(3 + 3) |(1,-1),(2,3)|`

= 3 + 2

= 5

∴ Adj A `=  [(C_11,C_12,C_13),(C_21,C_22,C_23),(C_31,C_32,C_33)]^T`

= `[(3,-12,6),(1,5,2),(-11,-1,5)]^T`

= `[(3,1,-11),(-12,5,-1),(6,2,5)]`

shaalaa.com
  Is there an error in this question or solution?
Chapter 4: Determinants - Exercise 4.5 [Page 131]

APPEARS IN

NCERT Mathematics Part 1 and 2 [English] Class 12
Chapter 4 Determinants
Exercise 4.5 | Q 2 | Page 131

RELATED QUESTIONS

If A = `[(3,1),(-1,2)]` show that A2 – 5A + 7I = 0. Hence, find A–1.


If A is an invertible matrix of order 2, then det (A−1) is equal to ______.


Find the adjoint of the following matrix:
\[\begin{bmatrix}\cos \alpha & \sin \alpha \\ \sin \alpha & \cos \alpha\end{bmatrix}\]

Verify that (adj A) A = |A| I = A (adj A) for the above matrix.

Compute the adjoint of the following matrix:

\[\begin{bmatrix}2 & - 1 & 3 \\ 4 & 2 & 5 \\ 0 & 4 & - 1\end{bmatrix}\]

Verify that (adj A) A = |A| I = A (adj A) for the above matrix.


If \[A = \begin{bmatrix}- 1 & - 2 & - 2 \\ 2 & 1 & - 2 \\ 2 & - 2 & 1\end{bmatrix}\] , show that adj A = 3AT.


Find the inverse of the following matrix.

\[\begin{bmatrix}0 & 1 & - 1 \\ 4 & - 3 & 4 \\ 3 & - 3 & 4\end{bmatrix}\]

Find the inverse of the following matrix.

\[\begin{bmatrix}1 & 0 & 0 \\ 0 & \cos \alpha & \sin \alpha \\ 0 & \sin \alpha & - \cos \alpha\end{bmatrix}\]

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

\[A = \begin{bmatrix}- 8 & 5 \\ 2 & 4\end{bmatrix}\] satisfies the equation \[A^2 + 4A - 42I = O\]. Hence, find A−1.

If \[A = \begin{bmatrix}3 & - 2 \\ 4 & - 2\end{bmatrix}\], find the value of \[\lambda\]  so that \[A^2 = \lambda A - 2I\]. Hence, find A−1.


If \[A = \begin{bmatrix}2 & - 1 & 1 \\ - 1 & 2 & - 1 \\ 1 & - 1 & 2\end{bmatrix}\].
Verify that \[A^3 - 6 A^2 + 9A - 4I = O\]  and hence find A−1.

If \[A = \frac{1}{9}\begin{bmatrix}- 8 & 1 & 4 \\ 4 & 4 & 7 \\ 1 & - 8 & 4\end{bmatrix}\],
prove that  \[A^{- 1} = A^3\]

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 

\[\begin{bmatrix}3 & 2 \\ 7 & 5\end{bmatrix} X \begin{bmatrix}- 1 & 1 \\ - 2 & 1\end{bmatrix} = \begin{bmatrix}2 & - 1 \\ 0 & 4\end{bmatrix}\]

 


Find the matrix X satisfying the equation 

\[\begin{bmatrix}2 & 1 \\ 5 & 3\end{bmatrix} X \begin{bmatrix}5 & 3 \\ 3 & 2\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} .\]

Find the inverse by using elementary row transformations:

\[\begin{bmatrix}1 & 6 \\ - 3 & 5\end{bmatrix}\]


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:

\[\begin{bmatrix}3 & - 3 & 4 \\ 2 & - 3 & 4 \\ 0 & - 1 & 1\end{bmatrix}\]


Find the inverse by using elementary row transformations:

\[\begin{bmatrix}2 & - 1 & 3 \\ 1 & 2 & 4 \\ 3 & 1 & 1\end{bmatrix}\]


If A is an invertible matrix such that |A−1| = 2, find the value of |A|.


If \[A = \begin{bmatrix}1 & - 3 \\ 2 & 0\end{bmatrix}\], write adj A.


If \[A = \begin{bmatrix}2 & 3 \\ 5 & - 2\end{bmatrix}\] , write  \[A^{- 1}\] in terms of A.


If A satisfies the equation \[x^3 - 5 x^2 + 4x + \lambda = 0\] then A-1 exists if _____________ .


The matrix \[\begin{bmatrix}5 & 10 & 3 \\ - 2 & - 4 & 6 \\ - 1 & - 2 & b\end{bmatrix}\] is a singular matrix, if the value of b is _____________ .


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 is an invertible matrix, then det (A1) is equal to ____________ .


If \[A = \begin{bmatrix}2 & - 1 \\ 3 & - 2\end{bmatrix},\text{ then } A^n =\] ______________ .

Using matrix method, solve the following system of equations: 
x – 2y = 10, 2x + y + 3z = 8 and -2y + z = 7


|A–1| ≠ |A|–1, where A is non-singular matrix.


Find the adjoint of the matrix A `= [(1,2),(3,4)].`


For what value of x, matrix `[(6-"x", 4),(3-"x", 1)]` is a singular matrix?


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 ____________.


A and B are invertible matrices of the same order such that |(AB)-1| = 8, If |A| = 2, then |B| is ____________.


If A = `[(1/sqrt(5), 2/sqrt(5)),((-2)/sqrt(5), 1/sqrt(5))]`, B = `[(1, 0),(i, 1)]`, i = `sqrt(-1)` and Q = ATBA, then the inverse of the matrix A. Q2021 AT is equal to ______.


If for a square matrix A, A2 – A + I = 0, then A–1 equals ______.


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:

  1. Convert the given above situation into a matrix equation of the form AX = B. (1)
  2. Find | A |. (1)
  3. Find A–1. (2)
    OR
    Determine P = A2 – 5A. (2)

Given that A is a square matrix of order 3 and |A| = –2, then |adj(2A)| is equal to ______.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×