Educational

Is AB – BA Symmetric or Skew-Symmetric? Determine Whether AB − BA is Symmetric or Skew-Symmetric Given: \[ A^T = A,\quad B^T = B \] Step 1: Take Transpose \[ (AB – BA)^T = (AB)^T – (BA)^T \] Step 2: Use Property \[ (AB)^T = B^T A^T,\quad (BA)^T = A^T B^T \] \[ (AB – […]

Is A^n Skew-Symmetric (Odd n)? Determine Whether An is Skew-Symmetric (n Odd) Given: \[ A \text{ is skew-symmetric } \Rightarrow A^T = -A \] \[ n \text{ is odd} \] Step 1: Take Transpose \[ (A^n)^T = (A^T)^n \] Step 2: Substitute AT = -A \[ (A^n)^T = (-A)^n = (-1)^n A^n \] Step 3:

Is A^n Symmetric or Skew-Symmetric? Determine Whether An is Symmetric or Skew-Symmetric Given: \[ A \text{ is skew-symmetric } \Rightarrow A^T = -A \] \[ n \text{ is even} \] Step 1: Take Transpose \[ (A^n)^T = (A^T)^n \] Step 2: Substitute AT = -A \[ (A^n)^T = (-A)^n = (-1)^n A^n \] Step 3:

Is A^n Symmetric or Skew-Symmetric? Determine Whether An is Symmetric or Skew-Symmetric Given: \[ A \text{ is symmetric } \Rightarrow A^T = A \] Step 1: Take Transpose of An \[ (A^n)^T = (A^T)^n \] Step 2: Use Property \[ (A^n)^T = A^n \] Conclusion: \[ (A^n)^T = A^n \Rightarrow A^n \text{ is symmetric} \]

Find λ for (A^n)^T = λA^n Find λ such that (An)T = λAn Given: \[ A \text{ is skew-symmetric } \Rightarrow A^T = -A \] Step 1: Take Transpose of Power \[ (A^n)^T = (A^T)^n \] Step 2: Substitute AT = -A \[ (A^n)^T = (-A)^n \] \[ (A^n)^T = (-1)^n A^n \] Conclusion: \[

Is ABA^T Symmetric or Skew-Symmetric? Determine Whether ABAT is Symmetric or Skew-Symmetric Given: \[ B \text{ is symmetric } \Rightarrow B^T = B \] Step 1: Take Transpose \[ (ABA^T)^T = (A^T)^T B^T A^T \] Step 2: Use Properties \[ (A^T)^T = A,\quad B^T = B \] \[ (ABA^T)^T = ABA^T \] Conclusion: \[ (ABA^T)^T

Is ABA^T Symmetric or Skew-Symmetric? Determine Whether ABAT is Symmetric or Skew-Symmetric Given: \[ B \text{ is skew-symmetric } \Rightarrow B^T = -B \] Step 1: Take Transpose \[ (ABA^T)^T = (A^T)^T B^T A^T \] Step 2: Use Properties \[ (A^T)^T = A,\quad B^T = -B \] \[ (ABA^T)^T = A(-B)A^T = -ABA^T \] Conclusion:

Condition for AB to be Symmetric Condition for AB to be Symmetric Given: \[ A \text{ and } B \text{ are symmetric matrices} \Rightarrow A^T = A,\; B^T = B \] Step 1: Take Transpose of AB \[ (AB)^T = B^T A^T \] Step 2: Use Symmetry \[ (AB)^T = BA \] Step 3: Condition

Sum of Elements of Skew-Symmetric Matrix Find Sum of All Elements of a Skew-Symmetric Matrix Given: \[ A = [a_{ij}] \text{ is skew-symmetric} \] Property Used: \[ a_{ij} = -a_{ji} \] Step 1: Pair Elements \[ a_{ij} + a_{ji} = 0 \] Step 2: Diagonal Elements \[ a_{ii} = 0 \] Step 3: Total Sum

Sum of Diagonal Elements of Skew-Symmetric Matrix Find Sum of Diagonal Elements of a Skew-Symmetric Matrix Given: \[ A = [a_{ij}] \text{ is skew-symmetric} \] Property Used: \[ A^T = -A \Rightarrow a_{ij} = -a_{ji} \] Step 1: Consider Diagonal Elements \[ a_{ii} = -a_{ii} \] Step 2: Solve \[ 2a_{ii} = 0 \Rightarrow a_{ii}