Advertisements
Advertisements
प्रश्न
If A = `[(1, 2, 2),(2, 1, 2),(2, 2, 1)]`, Show that A2 – 4A is a scalar matrix
उत्तर
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
संबंधित प्रश्न
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 matrix X so that X`[(1,2,3),(4,5,6)]= [(-7,-8,-9),(2,4,6)]`
If A = `[(alpha, beta),(gamma, -alpha)]` is such that A2 = I then ______.
Given `A = [(2,-3),(-4,7)]` compute `A^(-1)` and show that `2A^(-1) = 9I - A`
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
In a certain city there are 30 colleges. Each college has 15 peons, 6 clerks, 1 typist and 1 section officer. Express the given information as a column matrix. Using scalar multiplication, find the total number of posts of each kind in all the colleges.
If A and B are square matrices of order 3 such that |A| = –1, |B| = 3, then find the value of |2AB|.
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.
Show that (A + A') is symmetric matrix, if `A = ((2,4),(3,5))`
If A is a square matrix of order 3 with |A| = 4 , then the write the value of |-2A| .
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:
`[9 sqrt(2) -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:
`[(3, 0, 0),(0, 5, 0),(0, 0, 1/3)]`
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:
`[(1, 0, 0),(0, 1, 0),(0, 0, 1)]`
Identify the following matrix is singular or non-singular?
`[("a", "b", "c"),("p", "q", "r"),(2"a" - "p", 2"b" - "q", 2"c" - "r")]`
Find k if the following matrix is singular:
`[(4, 3, 1),(7, "k", 1),(10, 9, 1)]`
If A = `[(7, 3, 1),(-2, -4, 1),(5, 9, 1)]`, Find (AT)T.
The following matrix, using its transpose state whether it is symmetric, skew-symmetric, or neither:
`[(2, 5, 1),(-5, 4, 6),(-1, -6, 3)]`
Construct the matrix A = [aij]3 × 3 where aij = i − j. State whether A is symmetric or skew-symmetric.
Answer the following question:
If A = diag [2 –3 –5], B = diag [4 –6 –3] and C = diag [–3 4 1] then find 2A + B – 5C
Answer the following question:
If A = `[(1, omega),(omega^2, 1)]`, B = `[(omega^2, 1),(1, omega)]`, where ω is a complex cube root of unity, then show that AB + BA + A −2B is a null matrix
If A = `[(6, 0),("p", "q")]` is a scalar matrix, then the values of p and q are ______ respectively.
State whether the following statement is True or False:
If A is non singular, then |A| = 0
If A = `[(3, 1),(-1, 2)]`, then prove that A2 – 5A + 7I = O, where I is unit matrix of order 2
The matrix A = `[(0, 0, 5),(0, 5, 0),(5, 0, 0)]` is a ______.
If a matrix A is both symmetric and skew-symmetric, then ____________.
`root(3)(4663) + 349` = ? ÷ 21.003
A diagonal matrix is said to be a scalar matrix if its diagonal elements are
The number of all possible matrices of order 3/3, with each entry 0 or 1 is
A diagonal matrix in which all diagonal elements are same, is called a ______ matrix.
The minimum number of zeros in an upper triangular matrix will be ______.
If A and B are square matrices of order 3 × 3 and |A| = –1, |B| = 3, then |3AB| equals ______.
Let A = `[(0, -2),(2, 0)]`. If M and N are two matrices given by M = `sum_(k = 1)^10 A^(2k)` and N = `sum_(k = 1)^10 A^(2k - 1)` then MN2 is ______.
If A = `[(5, x),(y, 0)]` and A = AT, where AT is the transpose of the matrix A, then ______.
