Advertisements
Advertisements
Question
If `A=[[0,0],[4,0]]` find `A^16`
Solution
\[Given: A = \begin{bmatrix}0 & 0 \\ 4 & 0\end{bmatrix}\]
\[Here, \]
\[ A^2 = AA\]
\[ \Rightarrow A^2 = \begin{bmatrix}0 & 0 \\ 4 & 0\end{bmatrix}\begin{bmatrix}0 & 0 \\ 4 & 0\end{bmatrix}\]
\[ \Rightarrow A^2 = \begin{bmatrix}0 + 0 & 0 + 0 \\ 0 + 0 & 0 + 0\end{bmatrix}\]
\[ \Rightarrow A^2 = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
\[ A^4 = A^2 A^2 \]
\[ \Rightarrow A^4 = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
\[ \Rightarrow A^4 = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
\[ A^8 = A^4 A^4 \]
\[ \Rightarrow A^8 = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
\[ \Rightarrow A^8 = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
\[ A^{16} = A^8 A^8 \]
\[ \Rightarrow A^{16} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
\[ \therefore A^{16} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
Thus, `A^16` is a null matrix .
APPEARS IN
RELATED QUESTIONS
Compute the indicated product:
`[(2,3,4),(3,4,5),(4,5,6)][(1,-3,5),(0,2,4), (3,0,5)]`
Show that AB ≠ BA in each of the following cases:
`A= [[5 -1],[6 7]]`And B =`[[2 1],[3 4]]`
Show that AB ≠ BA in each of the following cases:
`A = [[1,3,-1],[2,-1,-1],[3,0,-1]]` And `B= [[-2,3,-1],[-1,2,-1],[-6,9,-4]]`
If A = `[[0,c,-b],[-c,0,a],[b,-a,0]]`and B =`[[a^2 ,ab,ac],[ab,b^2,bc],[ac,bc,c^2]]`, show that AB = BA = O3×3.
For the following matrices verify the distributivity of matrix multiplication over matrix addition i.e. A (B + C) = AB + AC:
`A = [[1 -1],[0 2]] B= [[-1 0],[2 1]]`and `C= [[0 1],[1 -1]]`
If \[A = \begin{bmatrix}4 & - 1 & - 4 \\ 3 & 0 & - 4 \\ 3 & - 1 & - 3\end{bmatrix}\] , Show that A2 = I3.
If [x 4 1] `[[2 1 2],[1 0 2],[0 2 -4]]` `[[x],[4],[-1]]` = 0, find x.
If [1 −1 x] `[[0 1 -1],[2 1 3],[1 1 1]] [[0],[1],[1]]=`= 0, find x.
\[A = \begin{bmatrix}3 & 1 \\ - 1 & 2\end{bmatrix} and \text{ I} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]
If [1 1 x] `[[1 0 2],[0 2 1],[2 1 0]] [[1],[1],[1]]` = 0, find x.
Solve the matrix equations:
[2x 3] `[[1 2],[-3 0]] , [[x],[8]]=0`
If `A= [[1,2,0],[3,-4,5],[0,-1,3]]` compute A2 − 4A + 3I3.
If f (x) = x2 − 2x, find f (A), where A=
`A=[[3,2, 0],[1,4,0],[0,0,5]]` show that A2 − 7A + 10I3 = 0
Find the matrix A such that `=[[1,2,3],[4,5,6]]=[[-7,-8,-9],[2,4,6],[11,10,9]]`
If `P(x)=[[cos x,sin x],[-sin x,cos x]],` then show that `P(x),P(y)=P(x+y)=P(y)P(x).`
\[A = \begin{bmatrix}\cos \alpha + \sin \alpha & \sqrt{2}\sin \alpha \\ - \sqrt{2}\sin \alpha & \cos \alpha - \sin \alpha\end{bmatrix}\] ,prove that
\[A^n = \begin{bmatrix}\text{cos n α} + \text{sin n α} & \sqrt{2}\text{sin n α} \\ - \sqrt{2}\text{sin n α} & \text{cos n α} - \text{sin n α} \end{bmatrix}\] for all n ∈ N.
Give examples of matrices
A and B such that AB ≠ BA
Give examples of matrices
A and B such that AB = O but A ≠ 0, B ≠ 0.
If A and B are square matrices of the same order, explain, why in general
(A + B)2 ≠ A2 + 2AB + B2
Let `A=[[1,1,1],[3,3,3]],B=[[3,1],[5,2],[-2,4]]` and `C=[[4,2],[-3,5],[5,0]]`Verify that AB = AC though B ≠ C, A ≠ O.
A trust fund has Rs 30000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30000 among the two types of bonds. If the trust fund must obtain an annual total interest of(ii) Rs 2000
The monthly incomes of Aryan and Babban are in the ratio 3 : 4 and their monthly expenditures are in the ratio 5 : 7. If each saves ₹ 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value?
If `A= [[3],[5],[2]]` And B=[1 0 4] , Verify that `(AB)^T=B^TA^T`
Let `A= [[1,-1,0],[2,1,3],[1,2,1]]` And `B=[[1,2,3],[2,1,3],[0,1,1]]` Find `A^T,B^T` and verify that (A + B)T = AT + BT
If \[A = \begin{bmatrix}1 \\ 2 \\ 3\end{bmatrix}\] write AAT.
If \[A = \begin{bmatrix}- 3 & 0 \\ 0 & - 3\end{bmatrix}\] , find A4.
Write matrix A satisfying ` A+[[2 3],[-1 4]] =[[3 6],[- 3 8]]`.
If A is a matrix of order 3 × 4 and B is a matrix of order 4 × 3, find the order of the matrix of AB.
If A is a square matrix such that A2 = A, then write the value of 7A − (I + A)3, where I is the identity matrix.
If A and B are two matrices such n that AB = B and BA = A , `A^2 + B^2` is equal to
Let A = \[\begin{bmatrix}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{bmatrix}\], then An is equal to
If A and B are square matrices of the same order, then (A + B)(A − B) is equal to
If A, B and C are square matrices of same order, then AB = AC always implies that B = C
If A = `[(-3, -2, -4),(2, 1, 2),(2, 1, 3)]`, B = `[(1, 2, 0),(-2, -1, -2),(0, -1, 1)]` then find AB and use it to solve the following system of equations:
x – 2y = 3
2x – y – z = 2
–2y + z = 3
