Matrices & Determinants
Properties of matrices
Grade 12
Question:
<p>Let <em>A</em> and <em>B</em> be two \(2 \times 2\) matrices. Consider the statements:<br>(i) \(AB = O \Rightarrow A = O\) or \(B = O\)<br>(ii) \(AB = I_2 \Rightarrow A = B^{-1}\)<br>(iii) \((A+B)^2 = A^2 + 2AB + B^2\)<br>Then</p>
<p>(1) (i) and (ii) are false, (iii) is true</p>
<p>(2) (ii) and (iii) are false, (i) is true</p>
<p>(3) (i) is false, (ii) and (iii) are true</p>
<p>(4) (i) and (iii) are false, (ii) is true</p>
Step-by-Step Solution
Key Concept: Matrix multiplication is non-commutative and divisors of zero can exist in matrices—a product can be zero without either factor being zero. Also, matrix operations don't follow all the same algebraic rules as real numbers.
<p><strong>Step 1: Analyze Statement (i)</strong></p><p>AB = O ⇒ A = O or B = O?</p><p>This is FALSE. Counterexample: Let A = [1 0; 0 0] and B = [0 0; 0 1]. Then AB = O, but neither A nor B is O. Matrices are zero divisors.</p><p><strong>Step 2: Analyze Statement (ii)</strong></p><p>AB = I₂ ⇒ A = B⁻¹?</p><p>This is TRUE. If AB = I₂, then A is the left inverse of B. Multiplying both sides by B⁻¹ on the right: ABB⁻¹ = I₂B⁻¹ ⟹ A = B⁻¹. For square matrices, left and right inverses coincide.</p><p><strong>Step 3: Analyze Statement (iii)</strong></p><p>(A+B)² = A² + 2AB + B²?</p><p>This is FALSE. Expanding: (A+B)² = (A+B)(A+B) = A² + AB + BA + B². This equals A² + 2AB + B² only if AB = BA (commutative). Matrices generally don't commute.</p><p><strong>Step 4: Conclusion</strong></p><p>Only statement (ii) is true.</p><p>∴ Answer: D</p>
Correct Answer: D