Advertisements
Advertisements
Question
Write a 2 × 2 matrix which is both symmetric and skew-symmetric.
Solution
A matrix which is both symmetric and skew-symmetric is a null matrix.
Hence, the required matrix is
APPEARS IN
RELATED QUESTIONS
Compute the indicated product.
`[(1, -2),(2,3)][(1,2,3),(2,3,1)]`
Compute the indicated products:
`[[1 -2],[2 3]][[1 2 3],[-3 2 -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]]`
Compute the products AB and BA whichever exists in each of the following cases:
`A=[[3 2],[-1 0],[-1 1]]` and `B= [[4 5 6],[0 1 2]]`
Evaluate the following:
`([[1 3],[-1 -4]]+[[3 -2],[-1 1]])[[1 3 5],[2 4 6]]`
If A = `[[4 2],[-1 1]]`
, prove that (A − 2I) (A − 3I) = O
If A = `[[ cos 2θ sin 2θ],[ -sin 2θ cos 2θ]]`, find A2.
If A= `[[1 0 -2],[3 -1 0],[-2 1 1]]` B=,`[[0 5 -4],[-2 1 3],[-1 0 2]] and C=[[1 5 2],[-1 1 0],[0 -1 1]]` verify that A (B − C) = AB − AC.
\[A = \begin{bmatrix}2 & - 3 & - 5 \\ - 1 & 4 & 5 \\ 1 & - 3 & - 4\end{bmatrix}\] , Show that A2 = A.
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.
Solve the matrix equations:
`[1 2 1] [[1,2,0],[2,0,1],[1,0 ,2]][[0],[2],[x]]=0`
If `A= [[1,2,0],[3,-4,5],[0,-1,3]]` compute A2 − 4A + 3I3.
Find the matrix A such that [2 1 3 ] `[[-1,0,-1],[-1,1,0],[0,1,1]] [[1],[0],[-1]]=A`
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=[[1,1],[0,1]] ,` Prove that `A=[[1,n],[0,1]]` for all positive integers n.
Give examples of matrices
A and B such that AB = O but BA ≠ O.
If A and B are square matrices of the same order, explain, why in general
(A + B)2 ≠ A2 + 2AB + B2
If A and B are square matrices of the same order, explain, why in general
(A − B)2 ≠ A2 − 2AB + B2
To promote making of toilets for women, an organisation tried to generate awarness through (i) house calls, (ii) letters, and (iii) announcements. The cost for each mode per attempt is given below:
(i) ₹50 (ii) ₹20 (iii) ₹40
The number of attempts made in three villages X, Y and Z are given below:
(i) (ii) (iii)
X 400 300 100
Y 300 250 75
Z 500 400 150
Find the total cost incurred by the organisation for three villages separately, using matrices.
Let `A =[[2,-3],[-7,5]]` And `B=[[1,0],[2,-4]]` verify that
(2A)T = 2AT
If `A=[[-2],[4],[5]]` , B = [1 3 −6], verify that (AB)T = BT AT
write AB.
If \[A = \begin{bmatrix}\cos x & - \sin x \\ \sin x & \cos x\end{bmatrix}\] , find AAT
If \[A = \begin{bmatrix}\cos \alpha & - \sin \alpha \\ \sin \alpha & \cos \alpha\end{bmatrix}\] is identity matrix, then write the value of α.
If A, B are square matrices of order 3, A is non-singular and AB = O, then B is a
If S = [Sij] is a scalar matrix such that sij = k and A is a square matrix of the same order, then AS = SA = ?
If A = [aij] is a scalar matrix of order n × n such that aii = k, for all i, then trace of A is equal to
(a) nk (b) n + k (c) \[\frac{n}{k}\] (d) none of these
The matrix \[A = \begin{bmatrix}0 & 0 & 4 \\ 0 & 4 & 0 \\ 4 & 0 & 0\end{bmatrix}\] is a
If A and B are square matrices of the same order, then (A + B)(A − B) is equal to
Show that if A and B are square matrices such that AB = BA, then (A + B)2 = A2 + 2AB + B2.
Prove by Mathematical Induction that (A′)n = (An)′, where n ∈ N for any square matrix A.
If AB = BA for any two square matrices, prove by mathematical induction that (AB)n = AnBn
If A = `[(0, 1),(1, 0)]`, then A2 is equal to ______.
If matrix A = [aij]2×2, where aij `{:(= 1 "if i" ≠ "j"),(= 0 "if i" = "j"):}` then A2 is equal to ______.
If A and B are square matrices of the same order, then [k (A – B)]′ = ______.
If A, B and C are square matrices of same order, then AB = AC always implies that B = C
Three schools DPS, CVC, and KVS decided to organize a fair for collecting money for helping the flood victims. They sold handmade fans, mats, and plates from recycled material at a cost of Rs. 25, Rs.100, and Rs. 50 each respectively. The numbers of articles sold are given as
School/Article | DPS | CVC | KVS |
Handmade/fans | 40 | 25 | 35 |
Mats | 50 | 40 | 50 |
Plates | 20 | 30 | 40 |
Based on the information given above, answer the following questions:
- How many articles (in total) are sold by three schools?
Let a, b, c ∈ R be all non-zero and satisfy a3 + b3 + c3 = 2. If the matrix A = `((a, b, c),(b, c, a),(c, a, b))` satisfies ATA = I, then a value of abc can be ______.