Advertisements
Advertisements
Question
If A' = `[(3, 4),(-1, 2),(0, 1)]` and B = `[(-1, 2, 1),(1, 2, 3)]`, then verify that (A – B)' = A' – B'
Advertisements
Solution
We know that, A = `[(3, -1, 0),(4, 2, 1)]` and B' = `[(-1, 1),(2, 2),(1, 3)]`
Now, (A – B) = `[(3, -1, 0),(4, 2, 1)] - [(-1, 2, 1),(1, 2, 3)]`
= `[(3 + 1, -1 -2, 0 - 1),(4 - 1, 2 - 2, 1 - 3)]`
= `[(4, -3, -1),(3, 0, -2)]`
So, (A – B)' = `[(4, 3),(-3, 0),(-1, -2)]` ...(i)
Then, A' – B' = `[(3, 4),(-1, 2),(0, 1)] - [(-1, 1),(2, 2),(1, 3)]`
= `[(3 + 1, 4 - 1),(-1 - 2, 2 - 2), (0 - 1, 1 - 3)]`
= `[(4, 3),(-3, 0),(-1, -2)]` ...(ii)
Equations (i) and (ii) prove that,
(A – B)' = A' – B'
APPEARS IN
RELATED QUESTIONS
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' = `[(-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 = `[(sin α, cos α), (-cos α, sin α)]`, then verify that A'A = I
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(6, -2, 2),(-2, 3, -1),(2, -1, 3)]`
If A and B are symmetric matrices, prove that AB – BA is 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 = [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
Express the matrix A as the sum of a symmetric and a skew-symmetric matrix, where A = `[(2, 4, -6),(7, 3, 5),(1, -2, 4)]`
Let A = `[(2, 3),(-1, 2)]`. Then show that A2 – 4A + 7I = O. Using this result calculate A5 also.
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 `[(1, 0, 0),(0, 2, 0),(0, 0, 4)]` is a ______.
The matrix `[(0, -5, 8),(5, 0, 12),(-8, -12, 0)]` is a ______.
If A and B are matrices of same order, then (AB′ – BA′) is a ______.
______ matrix is both symmetric and skew-symmetric matrix.
Sum of two skew-symmetric matrices is always ______ matrix.
If A and B are symmetric matrices, then AB – BA is a ______.
If A and B are symmetric matrices of same order, then AB is symmetric if and only if ______.
AA′ is always a symmetric matrix for any matrix A.
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 ____________.
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 ______.
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 ______.
A square matrix \[A=[a_{ij}]_{n\times n}\] is symmetric when which condition holds?
A square matrix \[A=[a_{ij}]_{n\times n}\] is skew-symmetric when which condition holds?
For a skew-symmetric matrix \[A=[a_{ij}]_{n\times n}\], which relation holds for all \[i\] and \[j\]?
Which transpose property justifies the step \[(A+A^T)^T=A^T+(A^T)^T\]?
For \[B=\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix}\], what is its skew-symmetric part \[Q=\frac{1}{2}(B-B^T)\]?
For the matrices \[P=\frac{1}{2}(B+B^T)\] and \[Q=\frac{1}{2}(B-B^T)\], what is \[P+Q\]?
