Educational

AA^T Symmetric or Skew-Symmetric Determine Whether AAT is Symmetric or Skew-Symmetric To Prove: \[ AA^T \text{ is symmetric} \] Step 1: Take Transpose \[ (AA^T)^T = (A^T)^T A^T \] Step 2: Use Property \[ (A^T)^T = A \] \[ (AA^T)^T = AA^T \] Conclusion: \[ (AA^T)^T = AA^T \Rightarrow AA^T \text{ is symmetric} \] Hence, […]

Symmetric or Skew-Symmetric Matrix Determine Whether Matrix is Symmetric or Skew-Symmetric Given: \[ a_{ij} = i^2 – j^2 \] Step 1: Find aji \[ a_{ji} = j^2 – i^2 = -(i^2 – j^2) \] Step 2: Compare \[ a_{ji} = -a_{ij} \] Conclusion: \[ A^T = -A \] Hence, the matrix is skew-symmetric. Next Question

Find Matrix A Find Matrix A Given: \[ A + \begin{bmatrix} 2 & 3 \\ -1 & 4 \end{bmatrix} = \begin{bmatrix} 3 & -6 \\ -3 & 8 \end{bmatrix} \] Step 1: Rearrange \[ A = \begin{bmatrix} 3 & -6 \\ -3 & 8 \end{bmatrix} – \begin{bmatrix} 2 & 3 \\ -1 & 4 \end{bmatrix}

Form Matrix from aij = i + 2j Form the Matrix A Given: \[ a_{ij} = i + 2j,\quad A \text{ is } 2 \times 2 \] Step 1: Compute Elements \[ a_{11} = 1 + 2(1) = 3 \] \[ a_{12} = 1 + 2(2) = 5 \] \[ a_{21} = 2 + 2(1)

Find x from Matrix Equation Find x Given: \[ \begin{bmatrix} x & 2 \end{bmatrix} \begin{bmatrix} 3 \\ 4 \end{bmatrix} = 2 \] Step 1: Multiply \[ 3x + 2 \cdot 4 = 2 \] \[ 3x + 8 = 2 \] Step 2: Solve \[ 3x = -6 \Rightarrow x = -2 \] Final Answer:

Is AA^T Symmetric or Skew-Symmetric? Is AAT Symmetric or Skew-Symmetric? To Prove: \[ AA^T \text{ is symmetric} \] Step 1: Take Transpose \[ (AA^T)^T = (A^T)^T A^T \] Step 2: Use Property \[ (A^T)^T = A \] \[ (AA^T)^T = AA^T \] Conclusion: \[ (AA^T)^T = AA^T \Rightarrow AA^T \text{ is symmetric} \] Hence, for

Check Symmetric or Skew-Symmetric Matrix Determine Whether Matrix is Symmetric or Skew-Symmetric Given: \[ a_{ij} = i^2 – j^2 \] Step 1: Find aji \[ a_{ji} = j^2 – i^2 = -(i^2 – j^2) \] Step 2: Compare \[ a_{ji} = -a_{ij} \] Conclusion: \[ A^T = -A \] Hence, the matrix is skew-symmetric. Next

Find Matrix A from Equation Find Matrix A Given: \[ A + \begin{bmatrix} 2 & 3 \\ -1 & 4 \end{bmatrix} = \begin{bmatrix} 3 & -6 \\ -3 & 8 \end{bmatrix} \] Step 1: Use Matrix Subtraction \[ A = \begin{bmatrix} 3 & -6 \\ -3 & 8 \end{bmatrix} – \begin{bmatrix} 2 & 3 \\

Form Matrix using aij = i + 2j Form the Matrix A Given: \[ a_{ij} = i + 2j,\quad A \text{ is } 2 \times 2 \] Step 1: Compute Each Element \[ a_{11} = 1 + 2(1) = 3 \] \[ a_{12} = 1 + 2(2) = 5 \] \[ a_{21} = 2 +

Find x from Matrix Equation Find x Given: \[ \begin{bmatrix} x & 2 \end{bmatrix} \begin{bmatrix} 3 \\ 4 \end{bmatrix} = 2 \] Step 1: Perform Matrix Multiplication \[ x \cdot 3 + 2 \cdot 4 = 2 \] \[ 3x + 8 = 2 \] Step 2: Solve for x \[ 3x = 2 –