Advertisements
Advertisements
प्रश्न
If A = [aij] is a skew-symmetric matrix, then write the value of \[\sum_i \sum_j\] aij.
Advertisements
उत्तर
\[Given: A = \left[ a_{ij} \right] \text{is a skew symmetric matrix} . \]
\[ \Rightarrow a_{ij} = - a_{ji} \left[ \text{For all values of i}, j \right]\]
\[ \Rightarrow a_{ii} = - a_{ii} \left[ \text{For all values of i} \right]\]
\[ \Rightarrow a_{ij} = 0\]
\[Now, \]
\[ \sum^{}_i \sum^{}_j a_{ij} = a_{11} + a_{12} + a_{13} + . . . + a_{21} + a_{22} + a_{23} + . . . + a_{31} + a_{32} + a_{33 + . . .} \]
\[ = 0 + a_{12} + a_{13} + . . . - a_{12} + 0 + a_{23} + . . . - a_{13} - a_{23} + 0 + . . . \]
\[ = 0\]
\[Thus, \]
APPEARS IN
संबंधित प्रश्न
Solve the following matrix equation for x: `[x 1] [[1,0],[−2,0]]=0`
If A = `([cos alpha, sin alpha],[-sinalpha, cos alpha])` , find α satisfying 0 < α < `pi/r`when `A+A^T=sqrt2I_2` where AT is transpose of A.
If `A=([2,0,1],[2,1,3],[1,-1,0])` find A2 - 5A + 4I and hence find a matrix X such that A2 - 5A + 4I + X = 0
Let `A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]`
Find A + B
Compute the following:
`[(a,b),(-b, a)] + [(a,b),(b,a)]`
Compute the following:
`[(a^2+b^2, b^2+c^2),(a^2+c^2, a^2+b^2)] + [(2ab , 2bc),(-2ac, -2ab)]`
Compute the following sums:
`[[2 1 3],[0 3 5],[-1 2 5]]`+ `[[1 -2 3],[2 6 1],[0 -3 1]]`
Let A = `[[2,4],[3,2]]`, `B=[[1,3],[-2,5]]`and `c =[[-2,5],[3,4]]`.Find each of the following: 2A − 3B
Let A = `[[2,4],[3,2]]`, `B=[[1,3],[-2,5]]`and `c =[[-2,5],[3,4]]`.Find each of the following: B − 4C
Let A = `[[2,4],[3,2]]`, `B=[[1,3],[-2,5]]`and `c =[[-2,5],[3,4]]`.Find each of the following: 3A − 2B + 3C
If A =`[[2,3],[5,7]],B =` `[[-1,0 ,2],[3,4,1]]`,`C= [[-1,2,3],[2,1,0]]`find : A + B and B + C
If A =`[[2 3],[5 7]],B =` `[[-1 0 2],[3 4 1]]`,`C= [[-1 2 3],[2 1 0]]`find
2B + 3A and 3C − 4B
Let A = `[[-1 0 2],[3 1 4]]``B=[[0 -2 5],[1 -3 1]]``and C = [[1 -5 2],[6 0 -4 ]]`Compute2A2-3B +4C :
If A = diag (2 − 59), B = diag (11 − 4) and C = diag (−6 3 4), find
B + C − 2A
Find X if Y =`[[3 2],[1 4]]`and 2X + Y =`[[1 0],[-3 2]]`
Find matrices X and Y, if 2X − Y = `[[6 -6 0],[-4 2 1]]`and X + 2Y =`[[3 2 5],[-2 1 -7 ]]`
If A =`[[9 1],[7 8]],B=[[1 5],[7 12]]`find matrix C such that 5A + 3B + 2C is a null matrix.
If A = `[[2 -2],[4 2],[-5 1]],B=[[8 0],[4 -2],[3 6]]`
, find matrix X such that 2A + 3X = 5B.
If A = `[[1 -3 2],[2 0 2]]`and `B = [[2 -1 -1],[1 0 -1]]` find the matrix C such that A + B + C is
, find the matrix C such that A + B + C is zero matrix.
Find x, y satisfying the matrix equations
`[[X-Y 2 -2],[4 x 6]]+[[3 -2 2],[1 0 -1]]=[[ 6 0 0],[ 5 2x+y 5]]`
Find x, y satisfying the matrix equations
`[x y + 2 z-3 ] + [ y 4 5]=[4 9 12]`
Find x, y satisfying the matrix equations
`x[[2],[1]]+y[[3],[5]]+[[-8],[-11]]=0`
If 2 `[[3 4],[5 x]]+[[1 y],[0 1]]=[[7 0],[10 5]]` find x and y.
Find x, y, z and t, if
`3[[x y],[z t]]=[[x 6],[-1 2t]]+[[4 x+y],[z+t 3]]`
Find x, y, z and t, if
`2[[x 5],[z t]]+[[x 6],[-1 2t]]=[[7 14],[15 14]]`
If X and Y are 2 × 2 matrices, then solve the following matrix equations for X and Y.
`2X + 3Y = [[2,3],[4,0]], 3X+2Y = [[-2,2],[1,-5]]`
Express the matrix \[A = \begin{bmatrix}3 & - 4 \\ 1 & - 1\end{bmatrix}\] as the sum of a symmetric and a skew-symmetric matrix.
If \[A = \begin{bmatrix}\cos x & \sin x \\ - \sin x & \cos x\end{bmatrix}\] , find x satisfying 0 < x < \[\frac{\pi}{2}\] when A + AT = I
If \[x\binom{2}{3} + y\binom{ - 1}{1} = \binom{10}{5}\] , find the value of x.
If \[2\begin{bmatrix}3 & 4 \\ 5 & x\end{bmatrix} + \begin{bmatrix}1 & y \\ 0 & 1\end{bmatrix} = \begin{bmatrix}7 & 0 \\ 10 & 5\end{bmatrix}\] , find x − y.
If \[\begin{bmatrix}xy & 4 \\ z + 6 & x + y\end{bmatrix} = \begin{bmatrix}8 & w \\ 0 & 6\end{bmatrix}\] , write the value of (x + y + z).
If \[\binom{x + y}{x - y} = \begin{bmatrix}2 & 1 \\ 4 & 3\end{bmatrix}\binom{1}{ - 2}\] , then write the value of (x, y).
The trace of the matrix \[A = \begin{bmatrix}1 & - 5 & 7 \\ 0 & 7 & 9 \\ 11 & 8 & 9\end{bmatrix}\], is
If possible, find the sum of the matrices A and B, where A = `[(sqrt(3), 1),(2, 3)]`, and B = `[(x, y, z),(a, "b", 6)]`
If A = `[(2, 1)]`, B = `[(5, 3, 4),(8, 7, 6)]` and C = `[(-1, 2, 1),(1, 0, 2)]`, verify that A(B + C) = (AB + AC).
If A = `[(1, 0, -1),(2, 1, 3 ),(0, 1, 1)]`, then verify that A2 + A = A(A + I), where I is 3 × 3 unit matrix.
If A = `[(1, 2),(4, 1),(5, 6)]` B = `[(1, 2),(6, 4),(7, 3)]`, then verify that: (2A + B)′ = 2A′ + B′
Let A = `[(1, 2),(-1, 3)]`, B = `[(4, 0),(1, 5)]`, C = `[(2, 0),(1, -2)]` and a = 4, b = –2. Show that: A + (B + C) = (A + B) + C
Let A = `[(1, 2),(-1, 3)]`, B = `[(4, 0),(1, 5)]`, C = `[(2, 0),(1, -2)]` and a = 4, b = –2. Show that: (a + b)B = aB + bB
If A = `[(0, -x),(x, 0)]`, B = `[(0, 1),(1, 0)]` and x2 = –1, then show that (A + B)2 = A2 + B2.
If A = `[(1, 2),(4, 1)]`, find A2 + 2A + 7I.
`"A" = [(1,-1),(2,-1)], "B" = [("x", 1),("y", -1)]` and (A + B)2 = A2 + B2, then x + y = ____________.
If A `= [(0,2),(2,0)],` then A2 is ____________.
If a2 + b2 + c2 = –2 and f(x) = `|(1 + a^2x, (1 + b^2)x, (1 + c^2)x),((1 + a^2)x, 1 + b^2x, (1 + c^2)x),((1 + a^2)x, (1 + b^2)x, (1 + c^2)x)|` then f(x) is a polynomial of degree ______.
Let A = `[(1, -1),(2, α)]` and B = `[(β, 1),(1, 0)]`, α, β ∈ R. Let α1 be the value of α which satisfies (A + B)2 = `A^2 + [(2, 2),(2, 2)]` and α2 be the value of α which satisfies (A + B)2 = B2 . Then |α1 – α2| is equal to ______.
