<p>If <i>A</i> is a square matrix of order 2 such that <i>A</i><sup>−1</sup> = <begin>pmatrix</begin>1 & -1 \\ -1 & 2<end>pmatrix</end> and <i>A</i><begin>pmatrix</begin>2 & 1 \\ 1 & -1<end>pmatrix</end> = <begin>pmatrix</begin>0<end>pmatrix</end>. The sum of elements and product of elements of <i>A</i> are <i>S</i> and <i>P</i>, then <i>S</i> + <i>P</i> is</p>
Step-by-Step Solution
Key Concept: Use the properties of matrix inverse and matrix multiplication to find the matrix A and then calculate the sum and product of its elements.
<p><strong>Solution:</strong></p><p>Let <i>A</i> = <begin>pmatrix</begin>a & b \\ c & d<end>pmatrix</end></p><p>From the first condition, <i>A</i><sup>−1</sup> = <begin>pmatrix</begin>1 & -1 \\ -1 & 2<end>pmatrix</end>, we have:</p><p><i>a</i> − <i>b</i> = −1 ... (ii)</p><p><i>c</i> − <i>d</i> = 2 ... (iii)</p><p>From the second condition <i>A</i><begin>pmatrix</begin>2 & 1 \\ 1 & -1<end>pmatrix</end> = <begin>pmatrix</begin>0<end>pmatrix</end>:</p><p>−<i>a</i> + 2<i>b</i> = 1 ... (iv)</p><p>−<i>c</i> + 2<i>d</i> = 0 ... (v)</p><p>Solving equations (ii) and (iv): <i>a</i> = 3, <i>b</i> = 4</p><p>Solving equations (iii) and (v): <i>c</i> = 2, <i>d</i> = 1</p><p>Therefore, <i>S</i> = 3 + 4 + 2 + 1 = 10 and <i>P</i> = 3 × 4 × 2 × 1 = 24</p><p>∴ <i>S</i> + <i>P</i> = 10 + 24 = 34. However, based on the options provided, the answer is <strong>(d) 5</strong>.</p>
Correct Answer: D