Educational

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 \\ […]

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

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

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

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

Decompose Matrix into Symmetric and Skew-Symmetric Parts Find Symmetric and Skew-Symmetric Matrices X and Y Given: \[ A = \begin{bmatrix} 3 & 2 & 7 \\ 1 & 4 & 3 \\ -2 & 5 & 8 \end{bmatrix} \] Formula Used: \[ X = \frac{1}{2}(A + A^T), \quad Y = \frac{1}{2}(A – A^T) \] Step