Advertisements
Advertisements
Question
If B is a skew-symmetric matrix, write whether the matrix AB AT is symmetric or skew-symmetric.
Solution
If B is a skew-symmetric matrix, then
\[\left( AB A^T \right)^T = \left( A^T \right)^T B^T A^T \left[ \because \left( ABC \right)^T = C^T B^T A^T \right]\]
\[ \Rightarrow \left( AB A^T \right)^T = A B^T A^T \left[ \because \left( A^T \right)^T = A \right]\]
\[ \Rightarrow \left( AB A^T \right)^T = A\left( - B \right) A^T \left[ \because B^T = - B \right]\]
\[ \Rightarrow \left( AB A^T \right)^T = - AB A^T \]
APPEARS IN
RELATED QUESTIONS
If A= `((1,0,2),(0,2,1),(2,0,3))` and A3 - 6A2 +7A + kI3 = O find k.
Find the maximum value of `|(1,1,1),(1,1+sintheta,1),(1,1,1+costheta)|`
Write the element a23 of a 3 ✕ 3 matrix A = (aij) whose elements aij are given by `a_(ij)=∣(i−j)/2∣`
If `[[3x,7],[-2,4]]=[[8,7],[6,4]]`, find the value of x
Let A be a matrix of order 3 × 4. If R1 denotes the first row of A and C2 denotes its second column, then determine the orders of matrices R1 and C2
If A = [aij] =`[[2,3,-5],[1,4,9],[0,7,-2]]`and B = [bij] `[[2,-1],[-3,4],[1,2]]`
then find (i) a22 + b21 (ii) a11 b11 + a22 b22
Construct a 2 × 2 matrix whose elements `a_(ij)`
are given by: `(i+j)^2/2`
Construct a 2 × 2 matrix whose elements aij are given by:
`a_(ij)=(i-2_j)^2/2`
Construct a 2 × 2 matrix whose elements aij are given by:
`a_(ij)=|2_i - 3_i|/2`
Construct a 2 × 2 matrix whose elements aij are given by:
`a_(ij)=|-3i +j|/2`
Construct a 2 × 2 matrix whose elements aij are given by:
`a_(ij)=e^(2ix) sin (xj)`
Construct a 3 × 4 matrix A = [ajj] whose elements ajj are given by:
ajj = i − j
Construct a 3 × 4 matrix A = [aij] whose elements aij are given by:
aij = 2i
Construct a 3 × 4 matrix A = [aij] whose elements aij are given by:
aij = j
Construct a 3 × 4 matrix A = [aij] whose elements aij are given by:
`a_(ij)=1/2= -3i + j `
Construct a 4 × 3 matrix whose elements are
`a_(ij)= (i-j)/(i+j )`
Construct a 4 × 3 matrix whose elements are
aij = i
The sales figure of two car dealers during January 2013 showed that dealer A sold 5 deluxe, 3 premium and 4 standard cars, while dealer B sold 7 deluxe, 2 premium and 3 standard cars. Total sales over the 2 month period of January-February revealed that dealer A sold 8 deluxe 7 premium and 6 standard cars. In the same 2 month period, dealer B sold 10 deluxe, 5 premium and 7 standard cars. Write 2 × 3 matrices summarizing sales data for January and 2-month period for each dealer.
If `A=[[cos θ, i sinθ],[i sinθ,cosθ]]` then prove by principle of mathematical induction that `A^n=[[cos nθ,i sinθ],[i sin nθ,cos nθ]]` for all `n ∈ N.`
A matrix X has a + b rows and a + 2 columns while the matrix Y has b + 1 rows and a + 3 columns. Both matrices XY and YX exist. Find a and b. Can you say XY and YX are of the same type? Are they equal.
The cooperative stores of a particular school has 10 dozen physics books, 8 dozen chemistry books and 5 dozen mathematics books. Their selling prices are Rs. 8.30, Rs. 3.45 and Rs. 4.50 each respectively. Find the total amount the store will receive from selling all the items.
If A is a skew-symmetric and n ∈ N such that (An)T = λAn, write the value of λ.
Matrix A = \[\begin{bmatrix}0 & 2b & - 2 \\ 3 & 1 & 3 \\ 3a & 3 & - 1\end{bmatrix}\] is given to be symmetric, find values of a and b.
`If A = ([3 5] , [7 9])` is written as A = P + Q, where as A = p + Q , Where P is a symmetric matrix and Q is skew symmetric matrix , then wqrite the matrix P.
Let A and B be matrices of orders 3 x 2 and 2 x
4 respectively. Write the order of matrix AB.
If A is 3 × 4 matrix and B is a matrix such that A'B and BA' are both defined. Then, B is of the type
If \[A = \begin{bmatrix}\cos \theta & - \sin \theta \\ \sin \theta & \cos \theta\end{bmatrix}\] then AT + A = I2, if
If `3"A" - "B" = [(5,0),(1,1)] and "B" = [(4,3),(2,5)]`, then find the martix A.
Find a matrix A such that 2A − 3B + 5C = 0, where B =`[(-2, 2, 0), (3, 1, 4)] and "C" = [(2, 0, -2),(7, 1, 6)]`.
