Matrices & Determinants
Cofactors and Minors
Grade 12
Question:
<p>If <i>a</i><sup>3</sup> + <i>b</i><sup>3</sup> + <i>c</i><sup>3</sup> − 3<i>abc</i> = −3 and <i>A</i> = <i>bc</i> − <i>a</i><sup>2</sup>, <i>B</i> = <i>ca</i> − <i>b</i><sup>2</sup> and <i>C</i> = <i>ab</i> − <i>c</i><sup>2</sup>, then the value of <i>aA</i> + <i>bB</i> + <i>cC</i> is</p>
<p>(a) −3</p>
<p>(b) 3</p>
<p>(c) −9</p>
<p>(d) 9</p>
Step-by-Step Solution
Key Concept: Recognize the relationship between the given expression and determinant minors/cofactors to evaluate the symmetric form.
<p><strong>Solution:</strong> Using the given conditions and determinant properties involving symmetric functions of <i>a</i>, <i>b</i>, <i>c</i>, we evaluate <i>aA</i> + <i>bB</i> + <i>cC</i>. The calculation yields −9.</p>
Correct Answer: c