Advertisements
Advertisements
प्रश्न
Find the values of x, y, z if the matrix A = `[(0, 2y, z),(x, y, -z),(x, -y, z)]` satisfy the equation A'A = I.
Advertisements
उत्तर
Here, A = `[(0, 2y, z),(x, y, -z),(y, -y, z)]`
⇒ A' = `[(0, x, x),(2y, y, -y),(z, -z, z)]`
∴ A'A = `[(0, x, x),(2y, y, -y),(z, -z, z)][(0, 2y, z),(x, y, -z),(x, -y, z)]`
= `[(1, 0, 0),(0, 1, 0),(0, 0, 1)]`
= `[(0 + x^2 + x^2, 0 + xy - xy, -xz + xz),(0 + yz - yx, 4y^2 + y^2 + y^2, 2yz - yz - yz),(0 - zx + 2x, 2yz - zy - zy, z^2 + z^2 + z^2)]`
= `[(1, 0, 0),(0, 1, 0),(0, 0, 1)]`
= `[(2x^2, 0, 0),(6, 6y^2, 0),(0, 0, 3z^2)] = [(1, 0, 0),(0, 1, 0),(0, 0, 1)]`
∴ `2x^2 = 1, x = ±1/sqrt(2),`
`6y^2 = 1, y = ±1/sqrt(6),`
3z2 = 1
∴ `z = ±1/sqrt(3)`
Hence, `x = ±1/sqrt(2), y = ±1/sqrt(6), z = ±1/sqrt(3)`.
APPEARS IN
संबंधित प्रश्न
If A= `((3,5),(7,9))`is written as A = P + Q, where P is a symmetric matrix and Q is skew symmetric matrix, then write the matrix P.
If A is a skew symmetric matric of order 3, then prove that det A = 0
If A' = `[(3, 4),(-1, 2),(0, 1)]` and B = `[(-1, 2, 1),(1, 2, 3)]`, then verify that (A – B)' = A' – B'
For the matrices A and B, verify that (AB)′ = B'A', where A = `[(1),(-4),(3)]`, B = `[(-1, 2, 1)]`
For the matrix A = `[(1, 5),(6, 7)]` verify that (A + A') is a symmetric matrix.
For the matrix A = `[(1, 5),(6, 7)]`, verify that (A – A') is a skew symmetric matrix.
Find `1/2` (A + A') and `1/2` (A – A'), when A = `[(0, a, b),(-a, 0, c),(-b, -c, 0)]`
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(3, 5),(1, -1)]`
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)]`
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(1, 5),(-1, 2)]`
Show that the matrix B'AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
Show that all the diagonal elements of a skew symmetric matrix are zero.
If A and B are symmetric matrices of the same order, write whether AB − BA is symmetric or skew-symmetric or neither of the two.
If \[A = \begin{bmatrix}1 & 2 \\ 0 & 3\end{bmatrix}\] is written as B + C, where B is a symmetric matrix and C is a skew-symmetric matrix, then B is equal to.
If A = [aij] is a square matrix of even order such that aij = i2 − j2, then
If A and B are two matrices of order 3 × m and 3 × n respectively and m = n, then the order of 5A − 2B is
Show that a matrix which is both symmetric and skew symmetric is a zero matrix.
If A and B are symmetric matrices of the same order, then (AB′ –BA′) is a ______.
If A = `[(0, 1),(1, 1)]` and B = `[(0, -1),(1, 0)]`, show that (A + B)(A – B) ≠ A2 – B2
If A = `[(cosalpha, sinalpha),(-sinalpha, cosalpha)]`, and A–1 = A′, find value of α
If A, B are square matrices of same order and B is a skew-symmetric matrix, show that A′BA is skew-symmetric.
If A is a symmetric matrix, then A3 is a ______ matrix.
If A is skew-symmetric, then kA is a ______. (k is any scalar)
If A and B are symmetric matrices, then AB – BA is a ______.
If A and B are symmetric matrices, then BA – 2AB is a ______.
If A is symmetric matrix, then B′AB is ______.
If each of the three matrices of the same order are symmetric, then their sum is a symmetric matrix.
AA′ is always a symmetric matrix for any matrix A.
If P is of order 2 x 3 and Q is of order 3 x 2, then PQ is of order ____________.
If A and B are symmetric matrices of the same order, then ____________.
If A = `[(3, "x" - 1),(2"x" + 3, "x" + 2)]` is a symmetric matrix, then x = ____________.
If A is any square matrix, then which of the following is skew-symmetric?
The diagonal elements of a skew symmetric matrix are ____________.
If ax4 + bx3 + cx2 + dx + e = `|(2x, x - 1, x + 1),(x + 1, x^2 - x, x - 1),(x - 1, x + 1, 3x)|`, then the value of e is ______.
Number of symmetric matrices of order 3 × 3 with each entry 1 or – 1 is ______.
The value of |A|, if A = `[(0, 2x - 1, sqrt(x)),(1 - 2x, 0, 2sqrt(x)),(-sqrt(x), -2sqrt(x), 0)]`, where x ∈ R+, is ______.
Which of the following is correct?
