Advertisements
Advertisements
Question
If A `= [(1,3),(3,4)]` and A2 − kA − 5I = 0, then the value of k is ______.
Options
5
3
7
9
Solution
If A `= [(1,3),(3,4)]` and A2 - KA - 5I = 0, then K = 5.
Explanation:
Given,
`A = [(1,3),(3,4)]`
A2 = A.A
`=[(1,3),(3,4)] [(1,3),(3,4)]`
`= [((1)(1)+(3)(3),(1)(3)+(3)( 4)),((3)(1) + ( 4)(3),(3)(3) + ( 4)( 4))]`
`= [(1+9,3+12),(3+12,9+16)]`
`= [(10,15),(15,25)]`
Now, A2 − KA − 5I = 0 [Given]
`[(10,15),(15,25)]-k[(1,3),(3,4)]-5[(1,0),(0,1)]=[(0,0),(0,0)]`
`[(10-k-5,15-3k-0),(15-3k-0,25-4k-5)]=[(0,0),(0,0)]`
`[(5-k,15-3k),(15-3k,20-4k)]=[(0,0),(0,0)]`
On comparing
5 − k = 0
∴ k = 5
OR 15 - 3k = 0
∴ k = 5
OR 20 - 4k = 0
Hence, k = 5
APPEARS IN
RELATED QUESTIONS
Which of the given values of x and y make the following pair of matrices equal?
`[(3x+7, 5),(y+1, 2-3x)] = [(0,y-2),(8,4)]`
Compute the indicated product.
`[(2,1),(3,2),(-1,1)][(1,0,1),(-1,2,1)]`
Evaluate the following:
`([[1 3],[-1 -4]]+[[3 -2],[-1 1]])[[1 3 5],[2 4 6]]`
For the following matrices verify the distributivity of matrix multiplication over matrix addition i.e. A (B + C) = AB + AC:
`A=[[2 -1],[1 1],[-1 2]]` `B=[[0 1],[1 1]]` C=`[[1 -1],[0 1]]`
Show that the matrix \[A = \begin{bmatrix}5 & 3 \\ 12 & 7\end{bmatrix}\] is root of the equation A2 − 12A − I = O
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 `P(x)=[[cos x,sin x],[-sin x,cos x]],` then show that `P(x),P(y)=P(x+y)=P(y)P(x).`
Let `A= [[1,1,1],[0,1,1],[0,0,1]]` Use the principle of mathematical introduction to show that `A^n [[1,n,n(n+1)//2],[0,1,1],[0,0,1]]` for every position integer n.
Give examples of matrices
A, B and C such that AB = AC but B ≠ C, A ≠ 0.
If A and B are square matrices of the same order, explain, why in general
(A + B)2 ≠ A2 + 2AB + B2
If A and B are square matrices of the same order such that AB = BA, then show that (A + B)2 = A2 + 2AB + B2.
If A is an m × n matrix and B is n × p matrix does AB exist? If yes, write its order.
Given an example of two non-zero 2 × 2 matrices A and B such that AB = O.
If \[A = \begin{bmatrix}\cos x & - \sin x \\ \sin x & \cos x\end{bmatrix}\] , find AAT
If \[A = \begin{bmatrix}- 3 & 0 \\ 0 & - 3\end{bmatrix}\] , find A4.
If A = [aij] is a square matrix such that aij = i2 − j2, then write whether A is symmetric or skew-symmetric.
If `A=[[i,0],[0,i ]]` , n ∈ N, then A4n equals
If A and B are two matrices such that AB = A and BA = B, then B2 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 \[\begin{bmatrix}\cos\frac{2\pi}{7} & - \sin\frac{2\pi}{7} \\ \sin\frac{2\pi}{7} & \cos\frac{2\pi}{7}\end{bmatrix}^k = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\] then the least positive integral value of k is _____________.
If A, B are square matrices of order 3, A is non-singular and AB = O, then B is a
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
Give an example of matrices A, B and C such that AB = AC, where A is nonzero matrix, but B ≠ C.
If A and B are square matrices of the same order, then (kA)′ = ______. (k is any scalar)
