Matrices & Determinants
Symmetric matrices
Grade Class 12
Question:
Let M be a 2 x 2 symmetric matrix with integer entries. Then M is invertible if<br>(A) the first column of M is the transpose of the second row of M<br>(B) the second row of M is the transpose of the first column of M<br>(C) M is a diagonal matrix with nonzero entries in the main diagonal<br>(D) the product of entries in the main diagonal is not the square of an integer
(A) the first column of M is the transpose of the second row of M
(B) the second row of M is the transpose of the first column of M
(C) M is a diagonal matrix with nonzero entries in the main diagonal
(D) the product of entries in the main diagonal is not the square of an integer
Step-by-Step Solution
Key Concept: A 2x2 symmetric matrix M = [[a, b], [b, c]] is invertible if det(M) = ac - b^2 is not equal to 0. For integer entries, this means ac is not equal to b^2.
Let M = [[a, b], [b, c]] where a, b, c are integers. M is symmetric. M is invertible if det(M) = ac - b^2 is not equal to 0. This implies ac is not equal to b^2. Option (D) states that the product of entries in the main diagonal (ac) is not the square of an integer (b^2), which is the condition for invertibility.
Correct Answer: 4