Advertisements
Advertisements
Question
If
Solution
\[Given: \hspace{0.167em} A = \begin{bmatrix}1 & 0 \\ - 1 & 7\end{bmatrix}\]
\[\]
\[Now, \]
\[ A^2 = AA\]
\[ \Rightarrow A^2 = \begin{bmatrix}1 & 0 \\ - 1 & 7\end{bmatrix}\begin{bmatrix}1 & 0 \\ - 1 & 7\end{bmatrix}\]
\[ \Rightarrow A^2 = \begin{bmatrix}1 - 0 & 0 + 0 \\ - 1 - 7 & 0 + 49\end{bmatrix}\]
\[ \Rightarrow A^2 = \begin{bmatrix}1 & 0 \\ - 8 & 49\end{bmatrix}\]
\[ A^2 - 8A + kI = 0\]
\[ \Rightarrow \begin{bmatrix}1 & 0 \\ - 8 & 49\end{bmatrix} - 8\begin{bmatrix}1 & 0 \\ - 1 & 7\end{bmatrix} + k\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} = 0\]
\[ \Rightarrow \begin{bmatrix}1 & 0 \\ - 8 & 49\end{bmatrix} - \begin{bmatrix}8 & 0 \\ - 8 & 56\end{bmatrix} + \begin{bmatrix}k & 0 \\ 0 & k\end{bmatrix} = 0\]
\[ \Rightarrow \begin{bmatrix}1 - 8 + k & 0 - 0 + 0 \\ - 8 + 8 + 0 & 49 - 56 + k\end{bmatrix} = 0\]
\[ \Rightarrow \begin{bmatrix}- 7 + k & 0 \\ 0 & - 7 + k\end{bmatrix} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
\[\]
The corresponding elements of two equal matrices are equal .
\[ \therefore - 7 + k = 0 \]
\[ \Rightarrow k = 7 \]
\[\]
APPEARS IN
RELATED QUESTIONS
Let `A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]`
Find BA
Compute the indicated product.
`[(1),(2),(3)] [2,3,4]`
Compute the indicated product.
`[(3,-1,3),(-1,0,2)][(2,-3),(1,0),(3,1)]`
Compute the indicated products:
`[[a b],[-b a]][[a -b],[b a]]`
Show that AB ≠ BA in each of the following cases:
`A= [[5 -1],[6 7]]`And B =`[[2 1],[3 4]]`
If A =
\[\begin{bmatrix}2 & - 3 & - 5 \\ - 1 & 4 & 5 \\ 1 & - 3 & - 4\end{bmatrix}\]and B =
\[\begin{bmatrix}- 1 & 3 & 5 \\ 1 & - 3 & - 5 \\ - 1 & 3 & 5\end{bmatrix}\] , show that AB = BA = O3×3.
If A = `[[0,c,-b],[-c,0,a],[b,-a,0]]`and B =`[[a^2 ,ab,ac],[ab,b^2,bc],[ac,bc,c^2]]`, show that AB = BA = O3×3.
If A =`[[2 -3 -5],[-1 4 5],[1 -3 -4]]` and B =`[[2 -2 -4],[-1 3 4],[1 2 -3]]`
, show that AB = A and BA = B.
\[A = \begin{bmatrix}3 & 1 \\ - 1 & 2\end{bmatrix}\]show that A2 − 5A + 7I = O use this to find A4.
If f (x) = x3 + 4x2 − x, find f (A), where\[A = \begin{bmatrix}0 & 1 & 2 \\ 2 & - 3 & 0 \\ 1 & - 1 & 0\end{bmatrix}\]
Give examples of matrices
A and B such that AB = O but A ≠ 0, B ≠ 0.
Give examples of matrices
A, B and C such that AB = AC but B ≠ C, A ≠ 0.
In a legislative assembly election, a political group hired a public relations firm to promote its candidates in three ways: telephone, house calls and letters. The cost per contact (in paise) is given matrix A as
Cost per contact
`A=[[40],[100],[50]]` `[["Teliphone"] ,["House call "],[" letter"]]`
The number of contacts of each type made in two cities X and Y is given in matrix B as
Telephone House call Letter
`B= [[ 1000, 500, 5000],[3000,1000, 10000 ]]`
Find the total amount spent by the group in the two cities X and Y.
A trust fund has Rs 30000 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 30000 among the two types of bonds. If the trust fund must obtain an annual total interest of
(i) Rs 1800
There are 2 families A and B. There are 4 men, 6 women and 2 children in family A, and 2 men, 2 women and 4 children in family B. The recommend daily amount of calories is 2400 for men, 1900 for women, 1800 for children and 45 grams of proteins for men, 55 grams for women and 33 grams for children. Represent the above information using matrix. Using matrix multiplication, calculate the total requirement of calories and proteins for each of the two families. What awareness can you create among people about the planned diet from this question?
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?
Let `A= [[1,-1,0],[2,1,3],[1,2,1]]` And `B=[[1,2,3],[2,1,3],[0,1,1]]` Find `A^T,B^T` and verify that (A + B)T = AT + BT
If `A=[[-2],[4],[5]]` , B = [1 3 −6], verify that (AB)T = BT AT
If \[A = \begin{bmatrix}- 3 & 0 \\ 0 & - 3\end{bmatrix}\] , find A4.
If A is a matrix of order 3 × 4 and B is a matrix of order 4 × 3, find the order of the matrix of AB.
If A is 2 × 3 matrix and B is a matrix such that AT B and BAT both are defined, then what is the order of B ?
If S = [Sij] is a scalar matrix such that sij = k and A is a square matrix of the same order, then AS = SA = ?
The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is
The number of possible matrices of order 3 × 3 with each entry 2 or 0 is
If A and B are square matrices of the same order, then (A + B)(A − B) is equal to
If \[A = \begin{pmatrix}\cos\alpha & - \sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1\end{pmatrix},\] ,find adj·A and verify that A(adj·A) = (adj·A)A = |A| I3.
If A = `[[3,9,0] ,[1,8,-2], [7,5,4]]` and B =`[[4,0,2],[7,1,4],[2,2,6]]` , then find the matrix `B'A'` .
Let A and B be square matrices of the order 3 × 3. Is (AB)2 = A2B2? Give reasons.
If A = `[(0, 1),(1, 0)]`, then A2 is equal to ______.
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:
- What is the total amount of money collected by all three schools DPS, CVC, and KVS?
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 ______.
