Question

Given matrix equation: \[ \begin{bmatrix} x - y & 2x + z \\ 2x - y & 3z + w \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix} \] Find the values of \( x, y, z, \) and \( w \).

Hint:

When solving matrix equations, compare corresponding elements to create a system of equations, then solve step by step using substitution.

Answer 1

Step 1: Set up element-wise equations from the given matrices: \[ x - y = -1, \quad 2x + z = 5, \quad 2x - y = 0, \quad 3z + w = 13. \]
Step 2: Solve for \( x \) and \( y \) using the equations \( x - y = -1 \) and \( 2x - y = 0 \). From \( 2x - y = 0 \), it follows that \( y = 2x \). Substituting \( y = 2x \) into \( x - y = -1 \) yields: \[ x - 2x = -1 \quad \Rightarrow \quad -x = -1 \quad \Rightarrow \quad x = 1. \] Consequently, \( y = 2(1) = 2 \). 
Step 3: Solve for \( z \) using the equation \( 2x + z = 5 \): \[ 2(1) + z = 5 \quad \Rightarrow \quad z = 3. \] 
Step 4: Solve for \( w \) using the equation \( 3z + w = 13 \): \[ 3(3) + w = 13 \quad \Rightarrow \quad 9 + w = 13 \quad \Rightarrow \quad w = 4. \] The solution is \( x = 1, y = 2, z = 3, w = 4 \).