Advertisements
Advertisements
प्रश्न
If A = `[(0, 1),(1, 1)]` and B = `[(0, -1),(1, 0)]`, show that (A + B)(A – B) ≠ A2 – B2
उत्तर
Given that A = `[(0, 1),(1, 1)]` and B = `[(0, -1),(1, 0)]`
A + B = `[(0, 1),(1, 1)] + [(0, -1),(1, 0)]`
⇒ A + B = `[(0 + 0, 1 - 1),(1 + 1, 1 + 0)]`
⇒ A + B = `[(0, 0),(2, 1)]`
A – B = `[(0, 1),(1, 1)] - [(0, -1),(1, 0)]`
⇒ A – B = `[(0 - 0, 1 + 1),(1 - 1, 1 - 0)]`
⇒ A – B = `[(0, 2),(0, 1)]`
∴ `("A" + "B") * ("A" – "B") = [(0, 0),(2, 1)],[(0, 2),(0, 1)]`
= `[(0 + 0, 0 + 0),(0 + 0, 4 + 1)]`
= `[(0, 0),(0, 5)]`
Now, R.H.S. = A2 – B2
= `"A" * "A" – "B" * "B"`
= `[(0, 1),(1, 1)][(0, 1),(1, 1)] - [(0,-1),(1, 0)][(0, -1),(1, 0)]`
= `[(0 +1,0 +1),(0 + 1, 1 + 1)] - [(0 - 1, 0 + 0),(0 + 0, -1 + 0)]`
= `[(1, 1),(1, 2)] - [(-1, 0),(0, -1)]`
= `[(1 + 1, 1 -0),(1 -0, 2 + 1)]`
= `[(2, 1),(1, 3)]`
Hence, `[(0, 0),(0, 5)] ≠ [(2, 10),(1, 3)]`
Hence, (A + B) . (A – B) ≠ A2 – B2
APPEARS IN
संबंधित प्रश्न
If A is a skew symmetric matric of order 3, then prove that det A = 0
If `A = [(-1,2,3),(5,7,9),(-2,1,1)] "and" B = [(-4,1,-5),(1,2,0),(1,3,1)]` then verify that (A+ B)' = A' + B'
if `A' [(3,4),(-1, 2),(0,1)] and B = [((-1,2,1),(1,2,3))]` then verify that (A + B)' = A' + B'
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:
`[(3,3,-1),(-2,-2,1),(-4,-5,2)]`
If A and B are symmetric matrices, prove that AB − BA is a skew symmetric matrix.
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.
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 matrix A is both symmetric and skew-symmetric, then
If A and B are symmetric matrices, then ABA is
If \[A = \begin{bmatrix}2 & 0 & - 3 \\ 4 & 3 & 1 \\ - 5 & 7 & 2\end{bmatrix}\] is expressed as the sum of a symmetric and skew-symmetric matrix, then the symmetric matrix is
If A and B are two skew-symmetric matrices of same order, then AB is symmetric matrix if ______.
Show that A′A and AA′ are both symmetric matrices for any matrix A.
If A = `[(cosalpha, sinalpha),(-sinalpha, cosalpha)]`, and A–1 = A′, find value of α
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, B are square matrices of same order and B is a skew-symmetric matrix, show that A′BA is skew-symmetric.
The matrix `[(0, -5, 8),(5, 0, 12),(-8, -12, 0)]` is a ______.
AA′ is always a symmetric matrix for any matrix A.
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 ______.
If `[(2, 0),(5, 4)]` = P + Q, where P is symmetric, and Q is a skew-symmetric matrix, then Q is equal to ______.
Number of symmetric matrices of order 3 × 3 with each entry 1 or – 1 is ______.
If A and B are symmetric matrices of the same order, then AB – BA is ______.
