Matrices & Determinants Question - None

Question

The sum of values of p for which the equations $x + y + z = 1$, $x + 2y + 4z = p$, and $x + 4y + 10z = p^2$ have a solution is ________.

Accepted Answer

Step 1: Write the augmented matrix and perform row operations.

$$\begin{bmatrix} 1 & 1 & 1 & | & 1 \ 1 & 2 & 4 & | & p \ 1 & 4 & 10 & | & p^2 \end{bmatrix}$$

Step 2: Reduce: R₂ → R₂ - R₁ and R₃ → R₃ - R₁:

$$\begin{bmatrix} 1 & 1 & 1 & | & 1 \ 0 & 1 & 3 & | & p-1 \ 0 & 3 & 9 & | & p^2-1 \end{bmatrix}$$

Step 3: Reduce: R₃ → R₃ - 3R₂:

$$\begin{bmatrix} 1 & 1 & 1 & | & 1 \ 0 & 1 & 3 & | & p-1 \ 0 & 0 & 0 & | & p^2-1-3(p-1) \end{bmatrix}$$

Step 4: Simplify the last entry: $p^2 - 1 - 3p + 3 = p^2 - 3p + 2 = (p-1)(p-2)$

Step 5: For consistency, the last row must not be [0 0 0 | non-zero]. This requires:

$(p-1)(p-2) = 0$

Thus $p = 1$ or $p = 2$

Step 6: Sum of values: $1 + 2 = 3$

∴ Answer: 3

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