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' = `[(-2, 3),(1, 2)]` and B = `[(-1, 0),(1, 2)]`, then find (A + 2B)'
For the matrices A and B, verify that (AB)′ = B'A', where A = `[(1),(-4),(3)]`, B = `[(-1, 2, 1)]`
For the matrices A and B, verify that (AB)′ = B'A' where A = `[(0),(1),(2)]`, B = `[(1, 5, 7)]`
If A = `[(cos α, sin α), (-sin α, cos α)]`, then verify that A' A = I
Show that the matrix A = `[(0, 1, -1),(-1, 0, 1),(1, -1, 0)]` is a skew symmetric matrix.
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.
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(6, -2, 2),(-2, 3, -1),(2, -1, 3)]`
Show that the matrix B'AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
If the matrix A is both symmetric and skew symmetric, then ______.
Show that all the diagonal elements of a skew symmetric matrix are zero.
Write a square matrix which is both symmetric as well as skew-symmetric.
For what value of x, is the matrix \[A = \begin{bmatrix}0 & 1 & - 2 \\ - 1 & 0 & 3 \\ x & - 3 & 0\end{bmatrix}\] a skew-symmetric matrix?
If A is a square matrix, then AA is a
If A and B are symmetric matrices, then ABA is
If the matrix `((6,-"x"^2),(2"x"-15 , 10))` is symmetric, find the value of x.
Let A = `[(2, 3),(-1, 2)]`. Then show that A2 – 4A + 7I = O. Using this result calculate A5 also.
If the matrix `[(0, "a", 3),(2, "b", -1),("c", 1, 0)]`, is a skew symmetric matrix, find the values of a, b and c.
If A and B are matrices of same order, then (AB′ – BA′) is a ______.
______ matrix is both symmetric and skew-symmetric matrix.
If A and B are symmetric matrices, then BA – 2AB is a ______.
If each of the three matrices of the same order are symmetric, then their sum is a symmetric matrix.
If A and B are any two matrices of the same order, then (AB)′ = A′B′.
If A is skew-symmetric matrix, then A2 is a symmetric matrix.
If A and B are symmetric matrices of the same order, then ____________.
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 `= [(6,8,5),(4,2,3),(9,7,1)]` is the sum of a symmetric matrix B and skew-symmetric matrix C, then B is ____________.
Let A = `[(2, 3),(a, 0)]`, a ∈ R be written as P + Q where P is a symmetric matrix and Q is skew-symmetric matrix. If det(Q) = 9, then the modulus of the sum of all possible values of determinant of P is equal to ______.
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 ______.
Let A and B be and two 3 × 3 matrices. If A is symmetric and B is skewsymmetric, then the matrix AB – BA is ______.
For what value of k the matrix `[(0, k),(-6, 0)]` is a skew symmetric matrix?
If A and B are symmetric matrices of the same order, then AB – BA is ______.
A square matrix \[A=[a_{ij}]_{n\times n}\] is skew-symmetric when which condition holds?
Which identity is used to replace \[(A^T)^T\] by \[A\] when proving that \[A+A^T\] is symmetric?
Which expression writes any square matrix \[A\] as the sum of a symmetric and a skew-symmetric matrix?
For \[B=\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix}\], what is its symmetric part \[P=\frac{1}{2}(B+B^T)\]?
For \[B=\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix}\], what is its skew-symmetric part \[Q=\frac{1}{2}(B-B^T)\]?
