Advertisements
Advertisements
प्रश्न
`A=[[1,0,-3],[2,1,3],[0,1,1]]`then verify that A2 + A = A(A + I), where I is the identity matrix.
उत्तर
\[A = \begin{bmatrix}1 & 0 & - 3 \\ 2 & 1 & 3 \\ 0 & 1 & 1\end{bmatrix}\]\[A^2 = \begin{bmatrix}1 & 0 & - 3 \\ 2 & 1 & 3 \\ 0 & 1 & 1\end{bmatrix}\begin{bmatrix}1 & 0 & - 3 \\ 2 & 1 & 3 \\ 0 & 1 & 1\end{bmatrix}\]
\[ = \begin{bmatrix}1 + 0 + 0 & 0 + 0 - 3 & - 3 + 0 - 3 \\ 2 + 2 + 0 & 0 + 1 + 3 & - 6 + 3 + 3 \\ 0 + 2 + 0 & 0 + 1 + 1 & 0 + 3 + 1\end{bmatrix}\]
\[ = \begin{bmatrix}1 & - 3 & - 6 \\ 4 & 4 & 0 \\ 2 & 2 & 4\end{bmatrix}\]
L. H . S
\[A^2 + A = \begin{bmatrix}1 & - 3 & - 6 \\ 4 & 4 & 0 \\ 2 & 2 & 4\end{bmatrix} + \begin{bmatrix}1 & 0 & - 3 \\ 2 & 1 & 3 \\ 0 & 1 & 1\end{bmatrix}\]
\[ = \begin{bmatrix}1 + 1 & - 3 + 0 & - 6 - 3 \\ 4 + 2 & 4 + 1 & 0 + 3 \\ 2 + 0 & 2 + 1 & 4 + 1\end{bmatrix}\]
\[ = \begin{bmatrix}2 & - 3 & - 9 \\ 6 & 5 & 3 \\ 2 & 3 & 5\end{bmatrix}\]
R. H . S
\[A + I = \begin{bmatrix}1 & 0 & - 3 \\ 2 & 1 & 3 \\ 0 & 1 & 1\end{bmatrix} + \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\]
\[ = \begin{bmatrix}1 + 1 & 0 + 0 & - 3 + 0 \\ 2 + 0 & 1 + 1 & 3 + 0 \\ 0 + 0 & 1 + 0 & 1 + 1\end{bmatrix}\]
\[ = \begin{bmatrix}2 & 0 & - 3 \\ 2 & 2 & 3 \\ 0 & 1 & 2\end{bmatrix}\]
\[\]
\[A\left( A + I \right) = \begin{bmatrix}1 & 0 & - 3 \\ 2 & 1 & 3 \\ 0 & 1 & 1\end{bmatrix}\begin{bmatrix}2 & 0 & - 3 \\ 2 & 2 & 3 \\ 0 & 1 & 2\end{bmatrix}\]
\[ = \begin{bmatrix}2 + 0 + 0 & 0 + 0 - 3 & - 3 + 0 - 6 \\ 4 + 2 + 0 & 0 + 2 + 3 & - 6 + 3 + 6 \\ 0 + 2 + 0 & 0 + 2 + 1 & 0 + 3 + 2\end{bmatrix}\]
\[ = \begin{bmatrix}2 & - 3 & - 9 \\ 6 & 5 & 3 \\ 2 & 3 & 5\end{bmatrix}\]
Therfore,LHS=RHS
Hence,`A^2+A=A(A+l)` is verified
APPEARS IN
संबंधित प्रश्न
Compute the indicated products
`[(2,3,4),(3,4,5),(4,5,6)][(1,-3,5),(0,2,4), (3,0,5)]`
A trust fund has Rs. 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs. 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of Rs 2,000.
Compute the indicated products:
`[[a b],[-b a]][[a -b],[b a]]`
Compute the products AB and BA whichever exists in each of the following cases:
A = [1 −1 2 3] and B=`[[0],[1],[3],[2]]`
Evaluate the following:
`[[],[1 2 3],[]]` `[[1 0 2],[2 0 1],[0 1 2]]` `[[2],[4],[6]]`
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 = `[[2 -1],[3 2]]` and B = `[[0 4],[-1 7]]`find 3A2 − 2B + I
For the following matrices verify the associativity of matrix multiplication i.e. (AB) C = A(BC):
`A =-[[1 2 0],[-1 0 1]]`,`B=[[1 0],[-1 2],[0 3]]` and C= `[[1],[-1]]`
\[A = \begin{bmatrix}2 & - 3 & - 5 \\ - 1 & 4 & 5 \\ 1 & - 3 & - 4\end{bmatrix}\] , Show that A2 = A.
If `[[2 3],[5 7]] [[1 -3],[-2 4]]-[[-4 6],[-9 x]]` find x.
If
If A=, find k such that A2 = kA − 2I2
If A=then find λ, μ so that A2 = λA + μI
Find the value of x for which the matrix product`[[2 0 7],[0 1 0],[1 -2 1]]` `[[-x 14x 7x],[0 1 0],[x -4x -2x]]`equal an identity matrix.
If `A= [[1,2,0],[3,-4,5],[0,-1,3]]` compute A2 − 4A + 3I3.
Find the matrix A such that `[[2,-1],[1,0],[-3,-4]]A` `=[[-1,-8,-10],[1,-2,-5],[9,22,15]]`
If\[A = \begin{bmatrix}a & b \\ 0 & 1\end{bmatrix}\], prove that\[A^n = \begin{bmatrix}a^n & b( a^n - 1)/a - 1 \\ 0 & 1\end{bmatrix}\] for every positive integer n .
If B, C are n rowed square matrices and if A = B + C, BC = CB, C2 = O, then show that for every n ∈ N, An+1 = Bn (B + (n + 1) C).
Let A and B be square matrices of the order 3 × 3. Is (AB)2 = A2 B2? Give reasons.
Let `A=[[1,1,1],[3,3,3]],B=[[3,1],[5,2],[-2,4]]` and `C=[[4,2],[-3,5],[5,0]]`Verify that AB = AC though B ≠ C, A ≠ O.
In a parliament election, a political party hired a public relations firm to promote its candidates in three ways − telephone, house calls and letters. The cost per contact (in paisa) is given in matrix A as
\[A = \begin{bmatrix}140 \\ 200 \\ 150\end{bmatrix}\begin{array} \text{Telephone}\\{\text{House calls }}\\ \text{Letters}\end{array}\]
The number of contacts of each type made in two cities X and Y is given in the matrix B as
\[\begin{array}"Telephone & House calls & Letters\end{array}\]
\[B = \begin{bmatrix}1000 & 500 & 5000 \\ 3000 & 1000 & 10000\end{bmatrix}\begin{array} \\City X \\ City Y\end{array}\]
Find the total amount spent by the party in the two cities.
What should one consider before casting his/her vote − party's promotional activity of their social activities?
If `A=[[-2],[4],[5]]` , B = [1 3 −6], verify that (AB)T = BT AT
write AB.
Write matrix A satisfying ` A+[[2 3],[-1 4]] =[[3 6],[- 3 8]]`.
If A is a square matrix such that A2 = A, then write the value of 7A − (I + A)3, where I is the identity matrix.
Write the number of all possible matrices of order 2 × 2 with each entry 1, 2 or 3.
If A and B are two matrices such that AB = A and BA = B, then B2 is equal to
If A = `[(3, 5)]`, B = `[(7, 3)]`, then find a non-zero matrix C such that AC = BC.
Show that if A and B are square matrices such that AB = BA, then (A + B)2 = A2 + 2AB + B2.
A matrix which is not a square matrix is called a ______ matrix.
If matrix AB = O, then A = O or B = O or both A and B are null matrices.
If A, B and C are square matrices of same order, then AB = AC always implies that B = C
A trust fund has Rs. 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of Rs. 1,800.
If A = `[(1, 1, 1),(0, 1, 1),(0, 0, 1)]` and M = A + A2 + A3 + .... + A20, then the sum of all the elements of the matrix M is equal to ______.
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 ______.
