Matrices & Determinants
Systems of Linear Equations
Grade 12
Question:
<p>Let <m>A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}</m>, if <m>u_1</m> and <m>u_2</m> are column matrices such that <m>Au_1 = 0</m> and <m>Au_2 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}</m>, then <m>u_1 + u_2</m> is</p>
<p>(a) null matrix</p>
<p>(b) <m>\begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix}</m></p>
<p>(c) <m>\begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}</m></p>
<p>(d) <m>\begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}</m></p>
Step-by-Step Solution
Key Concept: Solve two matrix equations Au₁ = 0 and Au₂ = e₁ to find the column vectors, then add them.
<p><strong>Step 1:</strong> From <m>Au_1 = 0</m>, solve for <m>u_1</m> by finding the null space of A.</p><p>Since A is lower triangular with 1's on diagonal, <m>u_1 = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}</m> is the only solution.</p><p><strong>Step 2:</strong> From <m>Au_2 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}</m>, solve the system:</p><p><m>u_2 = \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix}</m>.</p><p><strong>Step 3:</strong> <m>u_1 + u_2 = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} + \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix}</m>.</p><p>∴ Answer is (b).</p>
Correct Answer: B