Advertisements
Advertisements
Question
If \[A = \begin{bmatrix}1 & - 1 \\ 2 & - 1\end{bmatrix}, B = \begin{bmatrix}a & 1 \\ b & - 1\end{bmatrix}\]and (A + B)2 = A2 + B2, values of a and b are
Options
a = 4, b = 1
a = 1, b = 4
a = 0, b = 4
a = 2, b = 4
Solution
a = 1, b = 4
\[Here, \]
\[ \left( A + B \right)^2 = A^2 + B^2 \]
\[ \Rightarrow A^2 + AB + BA + B^2 = A^2 + B^2 \]
\[ \Rightarrow AB + BA = O\]
\[ \Rightarrow AB = - BA\]
\[ \Rightarrow \begin{bmatrix}1 & - 1 \\ 2 & - 1\end{bmatrix}\begin{bmatrix}a & 1 \\ b & - 1\end{bmatrix} = - \begin{bmatrix}a & 1 \\ b & - 1\end{bmatrix}\begin{bmatrix}1 & - 1 \\ 2 & - 1\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}a - b & 2 \\ 2a - b & 3\end{bmatrix} = - \begin{bmatrix}a + 2 & - a - 1 \\ b - 2 & - b + 1\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}a - b & 2 \\ 2a - b & 3\end{bmatrix} = \begin{bmatrix}- a - 2 & a + 1 \\ b + 2 & b - 1\end{bmatrix}\]
The corresponding elements of two equal matrices are equal .
\[ \Rightarrow a + 1 = \text{2 and b - 1} = 3\]
\[ \therefore a = \text{1 and b} = 4\]
APPEARS IN
RELATED QUESTIONS
Let `A = [(2,4),(3,2)] , B = [(1,3),(-2,5)], C = [(-2,5),(3,4)]`
Find AB
Show that AB ≠ BA in each of the following cases:
`A = [[1,3,-1],[2,-1,-1],[3,0,-1]]` And `B= [[-2,3,-1],[-1,2,-1],[-6,9,-4]]`
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
If A = `[[ cos 2θ sin 2θ],[ -sin 2θ cos 2θ]]`, find A2.
Let A =`[[-1 1 -1],[3 -3 3],[5 5 5]]`and B =`[[0 4 3],[1 -3 -3],[-1 4 4]]`
, compute A2 − B2.
If A= `[[1 0 -2],[3 -1 0],[-2 1 1]]` B=,`[[0 5 -4],[-2 1 3],[-1 0 2]] and C=[[1 5 2],[-1 1 0],[0 -1 1]]` verify that A (B − C) = AB − AC.
If [x 4 1] `[[2 1 2],[1 0 2],[0 2 -4]]` `[[x],[4],[-1]]` = 0, find x.
Show that the matrix \[A = \begin{bmatrix}5 & 3 \\ 12 & 7\end{bmatrix}\] is root of the equation A2 − 12A − I = O
If [1 1 x] `[[1 0 2],[0 2 1],[2 1 0]] [[1],[1],[1]]` = 0, find x.
If f (x) = x2 − 2x, find f (A), where A=
`A=[[1,2,2],[2,1,2],[2,2,1]]`, then prove that A2 − 4A − 5I = 0
`A=[[3,2, 0],[1,4,0],[0,0,5]]` show that A2 − 7A + 10I3 = 0
If `A=[[0,0],[4,0]]` find `A^16`
`A=[[3,-5],[-4,2]]` then find A2 − 5A − 14I. Hence, obtain A3
Give examples of matrices
A and B such that AB = O but A ≠ 0, B ≠ 0.
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
write AB.
If \[A = \begin{bmatrix}\cos x & - \sin x \\ \sin x & \cos x\end{bmatrix}\] , find AAT
Write a 2 × 2 matrix which is both symmetric and skew-symmetric.
If \[A = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & - 1\end{bmatrix}\] , then A2 is equal to ___________ .
If A and B are two matrices such n that AB = B and BA = A , `A^2 + B^2` is equal to
If \[A = \begin{bmatrix}1 & 2 & x \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} and B = \begin{bmatrix}1 & - 2 & y \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\] and AB = I3, then x + y equals
The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is
If A = [aij] is a scalar matrix of order n × n such that aii = k, for all i, then trace of A is equal to
If A is a square matrix such that A2 = I, then (A − I)3 + (A + I)3 − 7A is equal to
If X = `[(3, 1, -1),(5, -2, -3)]` and Y = `[(2, 1, -1),(7, 2, 4)]`, find 2X – 3Y
If X = `[(3, 1, -1),(5, -2, -3)]` and Y = `[(2, 1, -1),(7, 2, 4)]`, find A matrix Z such that X + Y + Z is a zero matrix
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 ______.
If A and B are square matrices of the same order, then (AB)′ = ______.
A square matrix where every element is unity is called an identity matrix.
If A, B and C are square matrices of same order, then AB = AC always implies that B = C
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 (in Rs.) collected by schools CVC and KVS?
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.
