Question

If \( P = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix} \) and \( Q = \begin{bmatrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5 \end{bmatrix} \), find \( (QP) \) and hence solve the following system of equations using matrices : \( x - y = 3, 2x + 3y + 4z = 17, y + 2z = 7 \)

Hint:

When a question says "hence", you MUST use the computed product to find the inverse.

Answer 1

Concept:

• Matrix inversion property: If \( QP = kI \), then \( P^{-1} = \frac{1}{k}Q \).

Step 1: Calculate \( QP \)
\[ QP = \begin{bmatrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5 \end{bmatrix} \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix} = \begin{bmatrix} 6 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 6 \end{bmatrix} = 6I \]

Step 2: Solve the system
System is \( PX = B \), where \( B = [3, 17, 7]^T \).
\[ X = P^{-1}B = \frac{1}{6}QB = \frac{1}{6} \begin{bmatrix} 6 + 34 - 28 \\ -12 + 34 \\ - 28 6 - 17 + 35 \end{bmatrix} = \begin{bmatrix} 2 \\ -1 \\ 4 \end{bmatrix} \]
So \( x = 2, y = -1, z = 4 \).