Advertisements
Advertisements
प्रश्न
Construct the matrix A = [aij]3 × 3 where aij = i − j. State whether A is symmetric or skew-symmetric.
Advertisements
उत्तर
A = [aij]3 × 3 = `[("a"_11, "a"_12, "a"_13),("a"_21, "a"_22, "a"_23),("a"_31, "a"_32, "a"_33)]`
Given that: aij = i − j
∴ a11 = 1 − 1 = 0,
a12 = 1 − 2 = − 1,
a13 = 1 − 3 = − 2,
a21 = 2 − 1 = 1,
a22 = 2 − 2 = 0,
a23 = 2 − 3 = − 1,
a31 = 3 − 1 = 2,
a32 = 3 − 2 = 1,
a33 = 3 − 3 = 0.
∴ A = `[(0, -1, -2),(1, 0, -1),(2, 1, 0)]`
∵ aij = − aij for all i and j
∵ A is a skew-symmetric matrix.
APPEARS IN
संबंधित प्रश्न
Find the value of x, y, and z from the following equation:
`[(x+y+z), (x+z), (y+z)] = [(9),(5),(7)]`
Find the value of a, b, c, and d from the equation:
`[(a-b, 2a+c),(2a-b, 3x+d)] = [(-1,5),(0,13)]`
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 is a square matrix such that A2 = A, then (I + A)3 – 7 A is equal to ______.
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
If A and B are square matrices of order 3 such that |A| = –1, |B| = 3, then find the value of |2AB|.
Show that a matrix A = `1/2[(sqrt2,-isqrt2,0),(isqrt2,-sqrt2,0),(0,0,2)]` is unitary.
If\[A = \begin{bmatrix}2 & 3 \\ 4 & 5\end{bmatrix}\]prove that A − AT is a skew-symmetric matrix.
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:
`[(3, -2, 4),(0, 0, -5),(0, 0, 0)]`
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:
`[(0, 4, 7),(-4, 0, -3),(-7, 3, 0)]`
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:
`[(10, -15, 27),(-15, 0, sqrt(34)),(27, sqrt(34), 5/3)]`
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:
`[(0, 0, 1),(0, 1, 0),(1, 0, 0)]`
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:
`[(7, 3),(-2, "k")]`
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:
`[(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 A is non singular, then |A| = 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
The matrix A = `[(0, 0, 5),(0, 5, 0),(5, 0, 0)]` is a ______.
Show by an example that for A ≠ O, B ≠ O, AB = O
If A = `[(0,0,0),(0,0,0),(0,1,0)]` then A is ____________.
If `[("a","b"),("c", "-a")]`is a square root of the 2 x 2 identity matrix, then a, b, c satisfy the relation ____________.
If A is a square matrix, then A – A’ is a ____________.
The matrix `[(0,5,-7),(-5,0,11),(7,-11,0)]` is ____________.
The matrix `[(0,-5,8),(5,0,12),(-8,-12,0)]` is a ____________.
If `[(1,2),(3,4)],` then A2 - 5A is equal to ____________.
A matrix is said to be a column matrix if it has
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
If A and B are square matrices of order 3 × 3 and |A| = –1, |B| = 3, then |3AB| equals ______.
Let A and B be 3 × 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the systems of linear equations (A2B2 – B2A2)X = O, where X is a 3 × 1 column matrix of unknown variables and O is a 3 × 1 null matrix, has ______.
If A = `[(0, -tan θ/2),(tan θ/2, 0)]` and (I2 + A) (I2 – A)–1 = `[(a, -b),(b, a)]` then 13(a2 + b2) is equal to ______.
If A is a square matrix of order 3, then |2A| is equal to ______.
Assertion: Let the matrices A = `((-3, 2),(-5, 4))` and B = `((4, -2),(5, -3))` be such that A100B = BA100
Reason: AB = BA implies AB = BA for all positive integers n.
