Find A + A^T Find A + AT Given: \[ A = \begin{bmatrix} 2 & 3 \\ 5 & 7 \end{bmatrix} \] Step 1: Find AT \[ A^T = \begin{bmatrix} 2 & 5 \\ 3 & 7 \end{bmatrix} \] Step 2: Compute A + AT \[ A + A^T = \begin{bmatrix} 2 & 3 \\ […]
If A = [[2, 3], [5, 7]], find A,+,A^T Read More »
Non-Zero Matrices with AB = O Example of Non-Zero Matrices A and B such that AB = O Example: \[ A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} \] Step: Multiply AB \[ AB = \begin{bmatrix} 1 & 1 \\
Give an example of two non-zero 2×2 matrices A and B such that AB = O. Read More »
Find AA^T Find AAT Given: \[ A = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \] Step 1: Find AT \[ A^T = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} \] Step 2: Multiply AAT \[ AA^T = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} = \begin{bmatrix} 1\cdot1
If A = [[1], [2], [3]], write AA^T Read More »
Find AB Matrix Multiplication Find AB Given: \[ A = \begin{bmatrix} 4 & 3 \\ 1 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} -4 \\ 3 \end{bmatrix} \] Step 1: Check Order \[ A \text{ is } 2 \times 2, \quad B \text{ is } 2 \times 1 \] \[ AB \text{ exists and will
If A = [[4, 3], [1, 2]] and B = [[-4], [3]], write AB. Read More »
Order of AB for m×n and n×p Matrices Does AB Exist? Find Its Order Given: \[ A \text{ is of order } m \times n, \quad B \text{ is of order } n \times p \] Condition for Existence of AB: Matrix multiplication \(AB\) exists if the number of columns of A equals the number
If A is an m×n matrix and B is n×p matrix does AB exist ? If yes, write its order. Read More »
Order of AB and BA Find the Order of AB and BA Given: \[ A = \begin{bmatrix} 2 & 1 & 4 \\ 4 & 1 & 5 \end{bmatrix} \] \[ B = \begin{bmatrix} 3 & -1 \\ 2 & 2 \\ 1 & 3 \end{bmatrix} \] Step 1: Determine Order of A and B
Matrix Decomposition into Symmetric and Skew-Symmetric Express Matrix as Sum of Symmetric and Skew-Symmetric Matrices Given: \[ A = \begin{bmatrix} 3 & -2 & -4 \\ 3 & -2 & -5 \\ -1 & 1 & 2 \end{bmatrix} \] Formula Used: \[ S = \frac{1}{2}(A + A^T), \quad K = \frac{1}{2}(A – A^T) \] Step
Express Matrix as Sum of Symmetric and Skew-Symmetric Express Matrix as Sum of Symmetric and Skew-Symmetric Matrices Given: \[ A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix} \] Formula Used: \[ \text{Symmetric part } (S) = \frac{1}{2}(A + A^T), \quad \text{Skew-symmetric part } (K) = \frac{1}{2}(A – A^T) \] Step 1: Find
Symmetric Matrix and Proof A + A^T Symmetric Matrix and Proof that A + AT is Symmetric Definition: A square matrix \(A\) is called symmetric if \[ A^T = A \] Given: \[ A = \begin{bmatrix} 2 & 4 \\ 5 & 6 \end{bmatrix} \] Step 1: Find AT \[ A^T = \begin{bmatrix} 2 &
Express Matrix as Sum of Symmetric and Skew-Symmetric Express Matrix as Sum of Symmetric and Skew-Symmetric Matrices Given: \[ A = \begin{bmatrix} 4 & 2 & -1 \\ 3 & 5 & 7 \\ 1 & -2 & 1 \end{bmatrix} \] Formula Used: \[ \text{Symmetric part } (S) = \frac{1}{2}(A + A^T), \quad \text{Skew-symmetric part