Finding λ (Lambda)

Solution:

Step 1: Multiply matrices

\[ \lambda \begin{bmatrix} 1 & 0 & 2 \\ 3 & 4 & 5 \end{bmatrix} = \begin{bmatrix} \lambda & 0 & 2\lambda \\ 3\lambda & 4\lambda & 5\lambda \end{bmatrix} \] \[ 2 \begin{bmatrix} 1 & 2 & 3 \\ -1 & -3 & 2 \end{bmatrix} = \begin{bmatrix} 2 & 4 & 6 \\ -2 & -6 & 4 \end{bmatrix} \]

Step 2: Add matrices

\[ = \begin{bmatrix} \lambda+2 & 4 & 2\lambda+6 \\ 3\lambda-2 & 4\lambda-6 & 5\lambda+4 \end{bmatrix} \]

Step 3: Compare corresponding elements

From (1,1): \[ \lambda + 2 = 4 \Rightarrow \lambda = 2 \] Check consistency: \[ 2\lambda + 6 = 10 \Rightarrow 2(2)+6=10 \ ✓ \] \[ 3\lambda – 2 = 4 \Rightarrow 6-2=4 \ ✓ \] \[ 4\lambda – 6 = 2 \Rightarrow 8-6=2 \ ✓ \] \[ 5\lambda + 4 = 14 \Rightarrow 10+4=14 \ ✓ \]

Final Answer:

\[ \boxed{\lambda = 2} \]