Advertisements
Advertisements
Question
If A = `[(1, 2, 2),(2, 1, 2),(2, 2, 1)]`, Show that A2 – 4A is a scalar matrix
Advertisements
Solution
A2 = A · A = `[(1, 2, 2),(2, 1, 2),(2, 2, 1)] [(1, 2, 2),(2, 1, 2),(2, 2, 1)]`
= `[(1 + 4 + 4, 2 + 2 + 4, 2 + 4 + 2),(2 + 2 + 4, 4 + 1 + 4, 4 + 2 + 2),(2 + 4 + 2, 4 + 2 + 2, 4 + 4 + 1)]`
= `[(9, 8, 8),(8, 9, 8),(8, 8, 9)]`
∴ A2 – 4A = `[(9, 8, 8),(8, 9, 8),(8, 8, 9)] - 4[(1, 2, 2),(2, 1, 2),(2, 2, 1)]`
= `[(9, 8, 8),(8, 9, 8),(8, 8, 9)] -[(4, 8, 8),(8, 4, 8),(8, 8, 4)]`
= `[(5, 0, 0),(0, 5, 0),(0, 0, 5)]`
which is a scalar matrix.
APPEARS IN
RELATED QUESTIONS
If A is a square matrix, such that A2=A, then write the value of 7A−(I+A)3, where I is an identity matrix.
Find the value of x, y, and z from the following equation:
`[(x+y, 2),(5+z, xy)] = [(6,2), (5,8)]`
if `A = [(0, -tan alpha/2), (tan alpha/2, 0)]` and I is the identity matrix of order 2, show that I + A = `(I -A)[(cos alpha, -sin alpha),(sin alpha, cos alpha)]`
Let A = `[(0,1),(0,0)]`show that (aI+bA)n = anI + nan-1 bA , where I is the identity matrix of order 2 and n ∈ N
if `A = [(3,-4),(1,-1)]` then prove A"=` [(1+2n, -4n),(n, 1-2n)]` where n is any positive integer
If A and B are square matrices of the same order such that AB = BA, then prove by induction that AB" = B"A. Further, prove that (AB)" = A"B" for all n ∈ N
If A = `[(alpha, beta),(gamma, -alpha)]` is such that A2 = I then ______.
Determine the product `[(-4,4,4),(-7,1,3),(5,-3,-1)][(1,-1,1),(1,-2,-2),(2,1,3)]` and use it to solve the system of equations x - y + z = 4, x- 2y- 2z = 9, 2x + y + 3z = 1.
Use product `[(1,-1,2),(0,2,-3),(3,-2,4)][(-2,0,1),(9,2,-3),(6,1,-2)]` to solve the system of equations x + 3z = 9, −x + 2y − 2z = 4, 2x − 3y + 4z = −3
Given two matrices A and B
`A = [(1,-2,3),(1,4,1),(1,-3, 2)] and B = [(11,-5,-14),(-1, -1,2),(-7,1,6)]`
find AB and use this result to solve the following system of equations:
x - 2y + 3z = 6, x + 4x + z = 12, x - 3y + 2z = 1
A coaching institute of English (subject) conducts classes in two batches I and II and fees for rich and poor children are different. In batch I, it has 20 poor and 5 rich children and total monthly collection is Rs 9,000, whereas in batch II, it has 5 poor and 25 rich children and total monthly collection is Rs 26,000. Using matrix method, find monthly fees paid by each child of two types. What values the coaching institute is inculcating in the society?
If 𝒙 = r cos θ and y= r sin θ prove that JJ-1=1.
Using coding matrix A=`[(2,1),(3,1)]` encode the message THE CROW FLIES AT MIDNIGHT.
Find the non-singular matrices P & Q such that PAQ is in normal form where`[(1,2,3,4),(2,1,4,3),(3,0,5,-10)]`
Show that (A + A') is symmetric matrix, if `A = ((2,4),(3,5))`
if `vec"a"= 2hat"i" + 3hat"j"+ hat"k", vec"b" = hat"i" -2hat"j" + hat"k" and vec"c" = -3hat"i" + hat"j" + 2hat"k", "find" [vec"a" vec"b" vec"c"]`
Choose the correct alternative.
The matrix `[(8, 0, 0),(0, 8, 0),(0, 0, 8)]` is _______
Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:
`[(5),(4),(-3)]`
Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:
`[(6, 0),(0, 6)]`
Identify the following matrix is singular or non-singular?
`[("a", "b", "c"),("p", "q", "r"),(2"a" - "p", 2"b" - "q", 2"c" - "r")]`
Identify the following matrix is singular or non-singular?
`[(5, 0, 5),(1, 99, 100),(6, 99, 105)]`
Find k if the following matrix is singular:
`[("k" - 1, 2, 3),(3, 1, 2),(1, -2, 4)]`
Find x, y, z If `[(0, -5"i", x),(y, 0, z),(3/2, -sqrt(2), 0)]` is a skew symmetric matrix.
The following matrix, using its transpose state whether it is symmetric, skew-symmetric, or neither:
`[(1, 2, -5),(2, -3, 4),(-5, 4, 9)]`
The following matrix, using its transpose state whether it is symmetric, skew-symmetric, or neither:
`[(0, 1 + 2"i", "i" - 2),(-1 - 2"i", 0, -7),(2 - "i", 7, 0)]`
State whether the following statement is True or False:
If `[(3, 0),(0, 2)][(x),(y)] = [(3),(2)]`, then x = 1 and y = – 1
If A and B are matrices of same order, then (3A –2B)′ is equal to______.
If two matrices A and B are of the same order, then 2A + B = B + 2A.
AB = AC ⇒ B = C for any three matrices of same order.
If A `= [("cos x", - "sin x"),("sin x", "cos x")]`, find AAT.
The matrix `[(0,-5,8),(5,0,12),(-8,-12,0)]` is a ____________.
A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an
If all the elements are zero, then matrix is said to be
Find X, If `[X - 5 - 1] [(1, 0, 2),(0, 2, 1),(2, 0, 3)][(x),(4),(1)] ` = 0
A diagonal matrix in which all diagonal elements are same, is called a ______ matrix.
How many matrices can be obtained by using one or more numbers from four given numbers?
If A and B are square matrices of order 3 × 3 and |A| = –1, |B| = 3, then |3AB| equals ______.
