Advertisements
Advertisements
Question
If AB = BA for any two square matrices, prove by mathematical induction that (AB)n = AnBn
Solution
Let P(n): (AB)n = AnBn
So, P(1): (AB)1 = A1B1
⇒ AB = AB
So, P(1) is true.
Let P(n) is true for some k ∈ N
So, P(k): (AB)k = AkBk, k ∈ N .....(i)
Now (AB)k+1 = (AB)k(AB) ....(Using (i))
= AkBk(AB)
= AkBk–1(BA)B
= AkBk–1(AB)B .....(As given AB = BA)
= AkBk–1AB2
= AkBk–2(BA)B2
= AkBk–2ABB2
= AkBk–2AB3
.......
.......
= Ak+1Bk+1
Thus P(1) is true and whenever P(k) is true P(k + 1) is true.
So, P(n) is true for all n ∈ N.
APPEARS IN
RELATED QUESTIONS
Let `A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]`
Find AB
Compute the indicated products:
`[[1 -2],[2 3]][[1 2 3],[-3 2 -1]]`
Compute the indicated product:
`[(2,3,4),(3,4,5),(4,5,6)][(1,-3,5),(0,2,4), (3,0,5)]`
If A = `[[1 0],[0 1]]`,B`[[1 0],[0 -1]]`
and C= `[[0 1],[1 0]]`
, then show that A2 = B2 = C2 = I2.
If A =
\[\begin{bmatrix}2 & - 3 & - 5 \\ - 1 & 4 & 5 \\ 1 & - 3 & - 4\end{bmatrix}\]and B =
\[\begin{bmatrix}- 1 & 3 & 5 \\ 1 & - 3 & - 5 \\ - 1 & 3 & 5\end{bmatrix}\] , show that AB = BA = O3×3.
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.
Compute the elements a43 and a22 of the matrix:`A=[[0 1 0],[2 0 2],[0 3 2],[4 0 4]]` `[[2 -1],[-3 2],[4 3]] [[0 1 -1 2 -2],[3 -3 4 -4 0]]`
If \[A = \begin{bmatrix}4 & - 1 & - 4 \\ 3 & 0 & - 4 \\ 3 & - 1 & - 3\end{bmatrix}\] , Show that A2 = I3.
\[A = \begin{bmatrix}3 & 1 \\ - 1 & 2\end{bmatrix}\]show that A2 − 5A + 7I = O use this to find A4.
Solve the matrix equations:
`[1 2 1] [[1,2,0],[2,0,1],[1,0 ,2]][[0],[2],[x]]=0`
If f (x) = x2 − 2x, find f (A), where A=
Find the matrix A such that [2 1 3 ] `[[-1,0,-1],[-1,1,0],[0,1,1]] [[1],[0],[-1]]=A`
Find a 2 × 2 matrix A such that `A=[[1,-2],[1,4]]=6l_2`
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.
Let A and B be square matrices of the same order. Does (A + B)2 = A2 + 2AB + B2 hold? If not, why?
If A and B are square matrices of the same order such that AB = BA, then show that (A + B)2 = A2 + 2AB + B2.
The cooperative stores of a particular school has 10 dozen physics books, 8 dozen chemistry books and 5 dozen mathematics books. Their selling prices are Rs. 8.30, Rs. 3.45 and Rs. 4.50 each respectively. Find the total amount the store will receive from selling all the items.
In a parliament election, a political party hired a public relations firm to promote its candidates in three ways − telephone, house calls and letters. The cost per contact (in paisa) is given in matrix A as
\[A = \begin{bmatrix}140 \\ 200 \\ 150\end{bmatrix}\begin{array} \text{Telephone}\\{\text{House calls }}\\ \text{Letters}\end{array}\]
The number of contacts of each type made in two cities X and Y is given in the matrix B as
\[\begin{array}"Telephone & House calls & Letters\end{array}\]
\[B = \begin{bmatrix}1000 & 500 & 5000 \\ 3000 & 1000 & 10000\end{bmatrix}\begin{array} \\City X \\ City Y\end{array}\]
Find the total amount spent by the party in the two cities.
What should one consider before casting his/her vote − party's promotional activity of their social activities?
If \[A = \begin{bmatrix}- 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1\end{bmatrix}\] , find A3.
Write the number of all possible matrices of order 2 × 2 with each entry 1, 2 or 3.
If \[A = \begin{bmatrix}2 & - 1 & 3 \\ - 4 & 5 & 1\end{bmatrix}\text{ and B }= \begin{bmatrix}2 & 3 \\ 4 & - 2 \\ 1 & 5\end{bmatrix}\] then
If X = `[(3, 1, -1),(5, -2, -3)]` and Y = `[(2, 1, -1),(7, 2, 4)]`, find X + Y
If X = `[(3, 1, -1),(5, -2, -3)]` and Y = `[(2, 1, -1),(7, 2, 4)]`, find 2X – 3Y
The matrix P = `[(0, 0, 4),(0, 4, 0),(4, 0, 0)]`is a ______.
If A and B are square matrices of the same order, then (kA)′ = ______. (k is any scalar)
If A and B are two square matrices of the same order, then AB = BA.
If A `= [(1,3),(3,4)]` and A2 − kA − 5I = 0, then the value of k is ______.
If A = `[(1, 1, 1),(0, 1, 1),(0, 0, 1)]` and M = A + A2 + A3 + .... + A20, then the sum of all the elements of the matrix M is equal to ______.
