Question

If $A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & -1 & 0 \\ 3 & 3 & -4\end{array}\right]$, $B=\left[\begin{array}{l}1 \\ 1 \\ 2\end{array}\right]$ and $X=\left[\begin{array}{l}x_1 \\ x_2 \\ x_3\end{array}\right]$ such that $AX = B$, then the value of $x_1 + x_2 + x_3 =$

• A. 4
• B. 5
• C. 6
• D. 3

Hint:

Always look for quick combinations of your structural rows! If you add 4 times equation (1) directly to equation (3), you get: $4(x_1 - x_2 + x_3) + (3x_1 + 3x_2 - 4x_3) = 4(1) + 2 \implies 7x_1 - x_2 = 6$. Subtracting 3 times equation (2) ($6x_1 - 3x_2 = 3$) from this helps isolate variables without executing a full formal row reduction matrix table!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given a matrix equation $AX = B$ representing a system of three linear equations with three variables ($x_1, x_2, x_3$). We need to find the sum of these variables ($x_1 + x_2 + x_3$).

Step 2: Key Formula or Approach:
Write out the matrix product explicitly as a system of linear equations: 1. Row 1 multiplication: $x_1 - x_2 + x_3 = 1$ 2. Row 2 multiplication: $2x_1 - x_2 = 1$ 3. Row 3 multiplication: $3x_1 + 3x_2 - 4x_3 = 2$ We can solve for the individual values using row reduction (Gaussian elimination) or substitution.

Step 3: Detailed Explanation:
Let's express the system of equations explicitly: $$x_1 - x_2 + x_3 = 1 \quad \text{--- (1)}$$ $$2x_1 - x_2 = 1 \implies x_2 = 2x_1 - 1 \quad \text{--- (2)}$$ $$3x_1 + 3x_2 - 4x_3 = 2 \quad \text{--- (3)}$$ 4. Isolate $x_3$ from equation (1) by substituting equation (2): $$x_3 = 1 - x_1 + x_2$$ $$x_3 = 1 - x_1 + (2x_1 - 1) = x_1$$ So, we have $x_3 = x_1$. 5. Substitute the expressions for $x_2$ and $x_3$ into equation (3) to solve for $x_1$: $$3x_1 + 3(2x_1 - 1) - 4(x_1) = 2$$ $$3x_1 + 6x_1 - 3 - 4x_1 = 2$$ $$5x_1 - 3 = 2 \implies 5x_1 = 5 \implies x_1 = 1$$ 6. Now find the remaining variables using our substitution equations: $x_3 = x_1 = 1$ $x_2 = 2(1) - 1 = 1$ 4. Calculate the required sum: $$\text{Sum} = x_1 + x_2 + x_3 = 1 + 1 + 1 = 3$$

Step 4: Final Answer:
The value of the sum $x_1 + x_2 + x_3$ is 3, which matches option (D).