Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p>The value of <span class="math">a</span> for which the following system of equations <span class="math">a^3 x + (a+1)^3 y + (a+2)^3 z = 0</span>, <span class="math">ax + (a+1)y + (a+2)z = 0</span>, <span class="math">x + y + z = 0</span> has a non-trivial solution is equal to</p>
<p>(a) 2</p>
<p>(b) 1</p>
<p>(c) 0</p>
<p>(d) -1</p>
Step-by-Step Solution
Key Concept: For a homogeneous system of linear equations to have a non-trivial solution, the determinant of the coefficient matrix must equal zero, making the rows linearly dependent.
<p>For a homogeneous system to have a non-trivial solution, the determinant of the coefficient matrix must be zero.</p><p><span class="math">\begin{vmatrix} a^3 & (a+1)^3 & (a+2)^3 \\ a & a+1 & a+2 \\ 1 & 1 & 1 \end{vmatrix} = 0</span></p><p>The third row is proportional to differences, and this determinant equals zero when the rows are linearly dependent. Testing the given options, <span class="math">a = -1</span> makes the system have a non-trivial solution.</p>
Correct Answer: d