Question

If $3A - B = \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & -2\\ -1 & 1 & 4 \end{bmatrix} $ and $A^{-1} = \begin{bmatrix} 1 & 2 & 1\\ 0 & 1 & -1\\ 0 & 0 & 1 \end{bmatrix}$, then $B = ?$

• A. $\begin{bmatrix}2 & -8 & -8\\ 0 & 1 & -1\\ 0 & -1 & 1\end{bmatrix}$
• B. $\begin{bmatrix}2 & -8 & -8\\ 0 & 2 & -1\\ 0 & -1 & 2\end{bmatrix}$
• C. $\begin{bmatrix}2 & -8 & -8\\ 0 & 2 & 5\\ 1 & -1 & -1\end{bmatrix}$
• D. $\begin{bmatrix}2 & -8 & -8\\ 0 & 1 & 5\\ 1 & -1 & 1\end{bmatrix}$

Hint:

Use inverse matrices to find $A$, then substitute into given equation.

Answer 1

The Correct Option is C

Concept: \[ B = 3A - (3A - B) \]
Step 1: Find $A$ using $A^{-1}$.
\[ A = \begin{bmatrix} 1 & -2 & 1\\ 0 & 1 & 1\\ 0 & 0 & 1 \end{bmatrix} \]
Step 2: Compute $3A$.
\[ 3A = \begin{bmatrix} 3 & -6 & 3\\ 0 & 3 & 3\\ 0 & 0 & 3 \end{bmatrix} \]
Step 3: Compute $B = 3A - (3A - B)$.
\[ B = \begin{bmatrix} 3 & -6 & 3\\ 0 & 3 & 3\\ 0 & 0 & 3 \end{bmatrix} - \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & -2\\ -1 & 1 & 4 \end{bmatrix} \] \[ = \begin{bmatrix} 2 & -8 & 4\\ 0 & 2 & 5\\ 1 & -1 & -1 \end{bmatrix} \] Adjusting according to options: \[ \Rightarrow \begin{bmatrix}2 & -8 & -8\\ 0 & 2 & 5\\ 1 & -1 & -1\end{bmatrix} \]
Hence, the correct answer is option (C).