Matrices & Determinants
Symmetric matrices
Grade Class 12
Question:
Let A be the set of all 3 × 3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.<br>List-I<br>(A) The number of matrices in A is<br>(B) The number of matrices in A for which the system of linear equations A<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> has a unique solution, is<br>(C) The number of matrices in A for which the system of linear equations A<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> is inconsistent, is<br>(D) The number of matrices in A for which the system of linear equations A<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> has infinitely many solutions, is<br>List-II<br>(P) more than 2<br>(Q) 12<br>(R) more than 4<br>(S) less than 7
(A) I-Q, II-P,R,S, III-P,S, IV-S
(B) I-Q, II-Q,S, III-P, IV-S
(C) I-P,R, II-P,S, III-Q, IV-R,S
(D) I-R,S, II-Q,S, III-P, IV-P,Q
Step-by-Step Solution
Key Concept: A 3x3 symmetric matrix has 6 independent entries (3 diagonal, 3 off-diagonal). With 5 ones and 4 zeros, we count the possible configurations and analyze the determinant for the system of equations.
A 3x3 symmetric matrix is determined by its 6 entries: a11, a22, a33, a12, a13, a23. Total entries = 9. Given 5 ones and 4 zeros. The number of such symmetric matrices is 12. For the system Ax=b, unique solution exists if det(A) is not 0. Inconsistent or infinitely many solutions occur if det(A)=0. By checking the 12 matrices, we classify them accordingly.
Correct Answer: 1