Question

Consider the following system of equations: \[ \begin{pmatrix} 1 & 3 & 2 2 & 2 & -3 4 & 4 & -6 2 & 5 & 2 \end{pmatrix} \begin{pmatrix} x y z \end{pmatrix} = \begin{pmatrix} 1 1 2 1 \end{pmatrix} \] The value of \( y^2 + z^2 \) is _______.

• A. 34
• B. 41
• C. 25
• D. 18

Hint:

In non-square matrices, always check for linear dependencies first. Recognizing that row 3 was a multiple of row 2 immediately reduced the complexity of the problem.

Answer 1

The Correct Option is A

Concept:
A system of linear equations represented by \( AX = B \) can be solved using various methods such as Gaussian elimination, Cramer's rule (if square), or by identifying dependencies between the equations. In an overdetermined system (more equations than variables), a solution only exists if the additional equations are consistent with the primary ones.

Step 1: Extract the equations and identify redundancies.
From the matrix multiplication, we obtain four linear equations:
• \( x + 3y + 2z = 1 \)
• \( 2x + 2y - 3z = 1 \)
• \( 4x + 4y - 6z = 2 \)
• \( 2x + 5y + 2z = 1 \) Observation: Equation (3) is simply \( 2 \times \) Equation (2), meaning it provides no new information and the system is consistent so far.

Step 2: Solve the system using elimination.
Subtracting Equation (2) from Equation (4): \[ (2x + 5y + 2z) - (2x + 2y - 3z) = 1 - 1 \] \[ 3y + 5z = 0 \quad \Rightarrow \quad y = -\frac{5z}{3} \quad \cdots (5) \] Now, substitute \( y \) into Equation (1): \[ x + 3\left(-\frac{5z}{3}\right) + 2z = 1 \quad \Rightarrow \quad x - 5z + 2z = 1 \quad \Rightarrow \quad x = 1 + 3z \quad \cdots (6) \] Substitute both \( x \) and \( y \) back into Equation (2): \[ 2(1 + 3z) + 2\left(-\frac{5z}{3}\right) - 3z = 1 \] \[ 2 + 6z - \frac{10z}{3} - 3z = 1 \quad \Rightarrow \quad 3z - \frac{10z}{3} = -1 \] \[ \frac{9z - 10z}{3} = -1 \quad \Rightarrow \quad -z = -3 \quad \Rightarrow \quad \mathbf{z = 3} \]

Step 3: Calculate \( y \) and the final expression.
Using \( z = 3 \) in Equation (5): \[ y = -\frac{5(3)}{3} = \mathbf{-5} \] The question asks for the value of \( y^2 + z^2 \): \[ y^2 + z^2 = (-5)^2 + (3)^2 = 25 + 9 = 34 \]