Educational

Find a12 from aij = i/j Find a12 Given: \[ a_{ij} = \frac{i}{j}, \quad A \text{ is } 2 \times 2 \] Step 1: Identify Indices \[ a_{12} = \frac{1}{2} \] Final Answer: \[ \boxed{\frac{1}{2}} \] Next Question / Full Exercise

Possible Orders of Matrix with 5 Elements Possible Orders of a Matrix with 5 Elements Step 1: Use Formula \[ \text{Number of elements} = \text{rows} \times \text{columns} \] \[ m \times n = 5 \] Step 2: Factorize 5 \[ 5 = 1 \times 5 = 5 \times 1 \] Step 3: Possible Orders \[

Find y from Matrix Equation Find y Given: \[ \begin{bmatrix} x & x – y \\ 2x + y & 7 \end{bmatrix} = \begin{bmatrix} 3 & 1 \\ 8 & 7 \end{bmatrix} \] Step 1: Compare Elements \[ x = 3 \] \[ x – y = 1 \] Step 2: Substitute x \[ 3

Number of 2×2 Matrices with 0 and 1 Total Number of 2×2 Matrices with Entries 0 and 1 Step 1: Understand the Matrix \[ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \] Step 2: Possible Values Each entry can be either 0 or 1. \[ \text{Choices for each entry} = 2 \] Step

Find Order of Matrix B Find the Order of Matrix B Given: \[ A \text{ is of order } 2 \times 3 \] Step 1: Find AT \[ A^T \text{ is of order } 3 \times 2 \] Step 2: Condition for ATB \[ A^T (3 \times 2) \cdot B \Rightarrow \text{columns of } A^T

Find Symmetric Matrix B Find Symmetric Matrix B Given: \[ A = \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix} \] Formula: \[ B = \frac{1}{2}(A + A^T) \] Step 1: Find AT \[ A^T = \begin{bmatrix} 1 & 0 \\ 2 & 3 \end{bmatrix} \] Step 2: Compute A + AT \[ A

Evaluate (I + A)^2 – 3A Evaluate (I + A)2 − 3A Given: \[ A^2 = A \] Step 1: Expand \[ (I + A)^2 = I^2 + IA + AI + A^2 \] \[ = I + A + A + A^2 \] \[ = I + 2A + A^2 \] Step 2: Use

Find k from Matrix Multiplication Find k Given: \[ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 3 & 1 \\ 2 & 5 \end{bmatrix} = \begin{bmatrix} 7 & 11 \\ k & 23 \end{bmatrix} \] Step 1: Multiply Matrices \[ AB = \begin{bmatrix} 1(3) + 2(2) & 1(1) + 2(5) \\ 3(3)

Find α when Matrix is Identity Find α such that Matrix is Identity Given: \[ A = \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix} \] \[ A = I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \] Step 1: Compare Elements \[ \cos \alpha = 1

Order of Matrix AB Find the Order of Matrix AB Given: \[ A \text{ is of order } 3 \times 4,\quad B \text{ is of order } 4 \times 3 \] Step 1: Check Multiplication Condition \[ \text{Number of columns of A} = \text{Number of rows of B} = 4 \] So, AB exists. Step