Question

If \( \begin{bmatrix} 1 & x & 1 \end{bmatrix} \begin{bmatrix} 1 & 3 & 2 \\ 0 & 5 & 1 \\ 0 & 2 & 0 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \\ x \end{bmatrix} = 0 \), then the values of \( x \) are:

• A. $1, 5$
• B. $-1, -5$
• C. $1, 6$
• D. $-1, -6$
• E. $3, 3$

Hint:

Always double-check the arithmetic of the middle term in the row-matrix multiplication, as it often contains the bulk of the variable terms.

Answer 1

The Correct Option is D

Concept: Matrix multiplication is associative. To solve for $x$, we perform the multiplication step-by-step to obtain a quadratic equation.
• Multiplying a $1 \times 3$ matrix by a $3 \times 3$ matrix results in a $1 \times 3$ matrix.
• The final product with a $3 \times 1$ matrix results in a $1 \times 1$ scalar equation.

Step 1: Multiply the first two matrices.
Let the row matrix be $R = \begin{bmatrix} 1 & x & 1 \end{bmatrix}$ and the square matrix be $M = \begin{bmatrix} 1 & 3 & 2
0 & 5 & 1
0 & 2 & 0 \end{bmatrix}$. \[ R \times M = \begin{bmatrix} (1\cdot1 + x\cdot0 + 1\cdot0) & (1\cdot3 + x\cdot5 + 1\cdot2) & (1\cdot2 + x\cdot1 + 1\cdot0) \end{bmatrix} \] \[ = \begin{bmatrix} 1 & 5x + 5 & x + 2 \end{bmatrix} \]

Step 2: Multiply the result by the column matrix.
Multiply $\begin{bmatrix} 1 & 5x + 5 & x + 2 \end{bmatrix}$ by $\begin{bmatrix} 1
1
x \end{bmatrix}$: \[ 1(1) + (5x + 5)(1) + (x + 2)(x) = 0 \] \[ 1 + 5x + 5 + x^2 + 2x = 0 \] \[ x^2 + 7x + 6 = 0 \]

Step 3: Solve the quadratic equation.
Factorizing $x^2 + 7x + 6 = 0$: \[ x^2 + 6x + x + 6 = 0 \Rightarrow x(x + 6) + 1(x + 6) = 0 \] \[ (x + 1)(x + 6) = 0 \] Thus, $x = -1$ and $x = -6$.