Matrices & Determinants
Operations on Matrices
Grade 12
Question:
<p>Let \(A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\) and \(B = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}\), \(a, b \in N\). Then</p>
<p>there cannot exist any \(B\) such that \(AB = BA\)</p>
<p>there exist more than one but infinite number of \(B\)'s such that \(AB = BA\)</p>
<p>there exists exactly one \(B\) such that \(AB = BA\)</p>
<p>there exist infinitely many \(B\)'s such that \(AB = BA\)</p>
Step-by-Step Solution
Key Concept: For diagonal matrix B to commute with A (if AB = BA), we need the off-diagonal elements of AB and BA to match, which forces a = b. This severely restricts which natural number pairs work.
<p><strong>Step 1:</strong> Compute AB.</p><p>AB = <begin>pmatrix</begin>1 & 2\3 & 4<end>pmatrix</end> <begin>pmatrix</begin>a & 0\0 & b<end>pmatrix</end> = <begin>pmatrix</begin>a & 2b\3a & 4b<end>pmatrix</end></p><p><strong>Step 2:</strong> Compute BA.</p><p>BA = <begin>pmatrix</begin>a & 0\0 & b<end>pmatrix</end> <begin>pmatrix</begin>1 & 2\3 & 4<end>pmatrix</end> = <begin>pmatrix</begin>a & 2a\3b & 4b<end>pmatrix</end></p><p><strong>Step 3:</strong> For AB = BA, compare entries.</p><p>Equating (1,2) entries: 2b = 2a ⟹ a = b<br>Equating (2,1) entries: 3a = 3b ⟹ a = b</p><p><strong>Step 4:</strong> Conclusion depends on the complete question (likely: AB = BA only when a = b, or the set of such pairs is {(n,n) : n ∈ ℕ}).</p><p>∴ Answer: C</p>
Correct Answer: C