Let R = { <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mn>3</mn></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mn>2</mn></mtd><mtd><mi>d</mi></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>5</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> : a, b, c, d ∈ {0, 3, 5, 7, 11, 13, 17, 19} }. Then the number of invertible matrices in R is
Step-by-Step Solution
Key Concept: A matrix is invertible if its determinant is non-zero. For the given 3x3 matrix, the determinant is calculated by expanding along the third row: det(M) = -5(ad - bc). For the matrix to be invertible, ad - bc must not be zero.
The matrix is M = [[a, 3, b], [c, 2, d], [0, 5, 0]]. The determinant is det(M) = -5(ad - bc). For M to be invertible, det(M) \neq 0, which means ad \neq bc. The set S = {0, 3, 5, 7, 11, 13, 17, 19} has 8 elements. The total number of ways to choose a, b, c, d is 8^4 = 4096. We need to subtract the cases where ad = bc. If ad = bc = 0, then (a=0 or d=0) and (b=0 or c=0). This leads to 3780 invertible matrices.
Correct Answer: 3780