Advertisements
Advertisements
Question
Prove by Mathematical Induction that (A′)n = (An)′, where n ∈ N for any square matrix A.
Solution
Let P(n): (A′)n = (An)′
∴ P(1): (A′) = (A)′
⇒ A' = A'
⇒ P(1) is true.
Now, let P(k) = (A')k = (Ak)'
Where k ∈ N
And P(k + 1): (A')k+1 = (A')kA'
= (A')k'A'
= (AAk)' .....(As (AB)' = B'A')
= (Ak+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
Compute the indicated products
`[(2,3,4),(3,4,5),(4,5,6)][(1,-3,5),(0,2,4), (3,0,5)]`
Compute the indicated product.
`[(3,-1,3),(-1,0,2)][(2,-3),(1,0),(3,1)]`
Show that AB ≠ BA in each of the following cases
`A=[[-1 1 0],[0 -1 1],[2 3 4]]` and =B `[[1 2 3], [0 1 0],[1 1 0]]`
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]]`
Evaluate the following:
`([[1 3],[-1 -4]]+[[3 -2],[-1 1]])[[1 3 5],[2 4 6]]`
Evaluate the following:
`[[1 -1],[0 2],[2 3]]` `([[1 0 2],[2 0 1]]-[[0 1 2],[1 0 2]])`
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 = `[[4 2],[-1 1]]`
, prove that (A − 2I) (A − 3I) = O
If A = `[[ab,b^2],[-a^2,-ab]]` , show that A2 = O
If A = `[[ cos 2θ sin 2θ],[ -sin 2θ cos 2θ]]`, find A2.
For the following matrices verify the associativity of matrix multiplication i.e. (AB) C = A(BC):
`A=[[4 2 3],[1 1 2],[3 0 1]]`=`B=[[1 -1 1],[0 1 2],[2 -1 1]]` and `C= [[1 2 -1],[3 0 1],[0 0 1]]`
\[A = \begin{bmatrix}3 & 1 \\ - 1 & 2\end{bmatrix} and \text{ I} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]
If
If A=, find k such that A2 = kA − 2I2
If `P=[[x,0,0],[0,y,0],[0,0,z]]` and `Q=[[a,0,0],[0,b,0],[0,0,c]]` prove that `PQ=[[xa,0,0],[0,yb,0],[0,0,zc]]=QP`
If A is an m × n matrix and B is n × p matrix does AB exist? If yes, write its order.
If \[A = \begin{bmatrix}- 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1\end{bmatrix}\] , find A3.
Write matrix A satisfying ` A+[[2 3],[-1 4]] =[[3 6],[- 3 8]]`.
If A = [aij] is a square matrix such that aij = i2 − j2, then write whether A is symmetric or skew-symmetric.
If `[2 1 3]([-1,0,-1],[-1,1,0],[0,1,1])([1],[0],[-1])=A` , then write the order of matrix A.
If A and B are two matrices such that AB = A and BA = B, then B2 is equal to
If \[A = \begin{bmatrix}1 & 2 & x \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} and B = \begin{bmatrix}1 & - 2 & y \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\] and AB = I3, then x + y equals
If A is a square matrix such that A2 = I, then (A − I)3 + (A + I)3 − 7A is equal to
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 A = `[(0, 1),(1, 0)]`, then A2 is equal to ______.
If A `= [(1,-2,1),(2,1,3)]` and B `= [(2,1),(3,2),(1,1)],` then (AB)T is equal
