Advertisements
Advertisements
प्रश्न
Express the matrix \[A = \begin{bmatrix}3 & - 4 \\ 1 & - 1\end{bmatrix}\] as the sum of a symmetric and a skew-symmetric matrix.
उत्तर
\[Given: A = \begin{bmatrix}3 & - 4 \\ 1 & - 1\end{bmatrix}\]
\[ A^T = \begin{bmatrix}3 & 1 \\ - 4 & - 1\end{bmatrix}\]
\[Let X = \frac{1}{2}\left( A + A^T \right) = \frac{1}{2}\left( \begin{bmatrix}3 & - 4 \\ 1 & - 1\end{bmatrix} + \begin{bmatrix}3 & 1 \\ - 4 & - 1\end{bmatrix} \right) = \begin{bmatrix}3 & \frac{- 3}{2} \\ \frac{- 3}{2} & - 1\end{bmatrix}\]
\[ X^T = \begin{bmatrix}3 & \frac{- 3}{2} \\ \frac{- 3}{2} & - 1\end{bmatrix}^T = \begin{bmatrix}3 & \frac{- 3}{2} \\ \frac{- 3}{2} & - 1\end{bmatrix} = X\]
\[Let Y = \frac{1}{2}\left( A - A^T \right) = \frac{1}{2}\left( \begin{bmatrix}3 & - 4 \\ 1 & - 1\end{bmatrix} - \begin{bmatrix}3 & 1 \\ - 4 & - 1\end{bmatrix} \right) = \begin{bmatrix}0 & \frac{- 5}{2} \\ \frac{5}{2} & 0\end{bmatrix}\]
\[ Y^T = \begin{bmatrix}0 & \frac{- 5}{2} \\ \frac{5}{2} & 0\end{bmatrix}^T = \begin{bmatrix}0 & \frac{5}{2} \\ \frac{- 5}{2} & 0\end{bmatrix} = - \begin{bmatrix}0 & \frac{- 5}{2} \\ \frac{5}{2} & 0\end{bmatrix} = Y\]
`"therefore X is a symmetric matrix and Y is a skew - symmetric matrix ."`
APPEARS IN
संबंधित प्रश्न
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:
`[(-1,4, -6),(8,5,16),(2,8,5)] + [(12,7,6),(8,0,5),(3,2,4)]`
Compute the following sums:
`[[3 -2],[1 4]]+ [[-2 4 ],[1 3]]`
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: B − 4C
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 = diag (2 − 59), B = diag (11 − 4) and C = diag (−6 3 4), find
B + C − 2A
Find matrices X and Y, if X + Y =`[[5 2],[0 9]]`
and X − Y = `[[3 6],[0 -1]]`
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 = `[[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 z-3 ] + [ y 4 5]=[4 9 12]`
Find the value of λ, a non-zero scalar, if λ
Find a matrix X such that 2A + B + X = O, where
If A = `[[8 0],[4 -2],[3 6]]` and B = `[[2 -2],[4 2],[-5 1]]`
, then find the matrix X of order 3 × 2 such that 2A + 3X = 5B.
Find x, y, z and t, if
`3[[x y],[z t]]=[[x 6],[-1 2t]]+[[4 x+y],[z+t 3]]`
If \[x\binom{2}{3} + y\binom{ - 1}{1} = \binom{10}{5}\] , find the value of x.
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).
The trace of the matrix \[A = \begin{bmatrix}1 & - 5 & 7 \\ 0 & 7 & 9 \\ 11 & 8 & 9\end{bmatrix}\], is
Addition of matrices is defined if order of the matrices is ______.
If A = `[(1, 2),(-2, 1)]`, B = `[(2, 3),(3, -4)]` and C = `[(1, 0),(-1, 0)]`, verify: 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)B = aB + bB
Matrix multiplication is ______ over addition.
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 ______.
