Advertisements
Advertisements
Question
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.
Advertisements
Solution
We have, A = `[(1, 0, -1),(2, 1, 3 ),(0, 1, 1)]`
∴ A2 = A · A
= `[(1, 0, -1),(2, 1, 3),(0, 1, 1)] [(1, 0, -1),(2, 1, 3),(0, 1, 1)]`
= `[(1 + 0 + 0, 0 + 0 - 1, -1 + 0 - 1),(2 + 2 + 0, 0 + 1 + 3, -2 + 3 + 3),(0 + 2 + 0, 0 + 1 + 1, 0 + 3 + 1)]`
= `[(1, -1, -2),(4, 4, 4),(2, 2, 4)]`
∴ A2 + A = `[(1, -1, -2),(4, 4, 4),(2, 2, 4)] + [(1, 0, -4),(2, 1, 3),(0, 1, 1)]`
= `[(2, -1, -3),(6, 5, 7),(2, 3, 5)]` ......(i)
Now, A + I = `[(1, 0, -1),(2, 1, 3),(0, 1, 1)] + [(1, 0, 0),(0, 1, 0),(0, 0, 1)]`
= `[(2, 0, -1),(2, 2, 3),(0, 1, 2)]`
So, A(A + I) = `[(1, 0, -1),(2, 1, 3),(0, 1, 1)] [(2, 0, -1),(2, 2, 3),(0, 1, 2)]`
= `[(2 + 0 + 0, 0 + 0 - 1, -1 + 0 - 2),(4 + 2 + 0, 0 + 2 + 3, -2 + 3 + 6),(0 + 2 + 0, 0 + 2 + 1, 0 + 3 + 2)]`
= `[(2, -1, -3),(6, 5, 7),(2, 3, 5)]` .....(iii)
From (i) and (ii)
We get A2 + A = A(A + I)
APPEARS IN
RELATED QUESTIONS
if `A=[[2,0,0],[0,2,0],[0,0,2]]` then A6= ......................
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:
`[(-1,4, -6),(8,5,16),(2,8,5)] + [(12,7,6),(8,0,5),(3,2,4)]`
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: 3A − C
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
If A = diag (2 − 59), B = diag (11 − 4) and C = diag (−6 3 4), find
2A + 3B − 5C
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 ]]`
f X − Y =`[[1 1 1],[1 1 0],[1 0 0]]` and X + Y = `[[3 5 1],[-1 1 1],[11 8 0]]`find X and Y.
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.
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]`
If 2 `[[3 4],[5 x]]+[[1 y],[0 1]]=[[7 0],[10 5]]` find x and y.
Find a matrix X such that 2A + B + X = O, where
`A= [[-1 2],[3 4]],B= [[3 -2],[1 5]]`
Find x, y, z and t, if
`2[[x 5],[z t]]+[[x 6],[-1 2t]]=[[7 14],[15 14]]`
If w is a complex cube root of unity, show that
`([[1 w w^2],[w w^2 1],[w^2 1 w]]+[[w w^2 1],[w^2 1 w],[w w^2 1]])[[1],[w],[w^2]]=[[0],[0],[0]]`
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 A = [aij] is a skew-symmetric matrix, then write the value of \[\sum_i \sum_j\] aij.
Find the values of x and y, if \[2\begin{bmatrix}1 & 3 \\ 0 & x\end{bmatrix} + \begin{bmatrix}y & 0 \\ 1 & 2\end{bmatrix} = \begin{bmatrix}5 & 6 \\ 1 & 8\end{bmatrix}\]
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 \[\binom{x + y}{x - y} = \begin{bmatrix}2 & 1 \\ 4 & 3\end{bmatrix}\binom{1}{ - 2}\] , then write the value of (x, y).
If \[I = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}, J = \begin{bmatrix}0 & 1 \\ - 1 & 0\end{bmatrix} and B = \begin{bmatrix}\cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{bmatrix}\] then B equals )
The trace of the matrix \[A = \begin{bmatrix}1 & - 5 & 7 \\ 0 & 7 & 9 \\ 11 & 8 & 9\end{bmatrix}\], 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 = `[(2, 1)]`, B = `[(5, 3, 4),(8, 7, 6)]` and C = `[(-1, 2, 1),(1, 0, 2)]`, verify that A(B + C) = (AB + AC).
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
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 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 ______.
