Question

Let M be 2 × 2 symmetric matrix with integer entries, then M is invertible if

• A. the first column of M is the transpose of second row of M
• B. the second row of M is the transpose of first column of M
• C. M is diagonal matrix with non-zero entries in the principal diagonal
• D. The product of entries in the principal diagonal of M is the product of entries in the other diagonal

Answer 1

The Correct Option is C

Let \(M\) be a 2 × 2 symmetric matrix with integer entries. A symmetric matrix has the form \(M = \begin{bmatrix} a & b \\ b & c \end{bmatrix}\), where a, b, and c are integers.

A matrix is invertible if its determinant is non-zero. The determinant of \(M\) is given by \(\det(M) = ac - b^2\).

We need to find the condition under which \(M\) is invertible, i.e., \(ac - b^2 \neq 0\).

1. The first column of M is the transpose of the second row of M. This means \(\begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} b \\ c \end{bmatrix}\), so \(a = b\) and \(b = c\). Therefore, \(a = b = c\). In this case, \(\det(M) = a^2 - a^2 = 0\), so \(M\) is not invertible.
2. The second row of M is the transpose of the first column of M. This statement simply defines what a symmetric matrix is. It doesn't tell us anything more about the relationship between a, b and c. So \(\det(M) = ac-b^2\), can be 0 or non-zero, hence we don't have sufficient information
3. M is a diagonal matrix with non-zero entries in the principal diagonal. A diagonal matrix has the form \(M = \begin{bmatrix} a & 0 \\ 0 & c \end{bmatrix}\), where \(a \neq 0\) and \(c \neq 0\). In this case, \(b = 0\), and \(\det(M) = ac - 0^2 = ac\). Since \(a \neq 0\) and \(c \neq 0\), \(ac \neq 0\), so \(M\) is invertible.
4. The product of entries in the principal diagonal of M is the product of entries in the other diagonal. This means \(ac = b^2\). Therefore, \(\det(M) = ac - b^2 = 0\), so \(M\) is not invertible.

Therefore, \(M\) is invertible if \(M\) is a diagonal matrix with non-zero entries in the principal diagonal.

Answer 2

A \( 2 \times 2 \) symmetric matrix \( M \) has the form: \[ M = \begin{pmatrix} a & b \\ b & d \end{pmatrix} \] For \( M \) to be invertible, its determinant must be non-zero. The determinant of \( M \) is given by: \[ \text{det}(M) = ad - b^2 \] For the matrix to be invertible, we require that the determinant is non-zero, i.e., \( ad - b^2 \neq 0 \).
This condition is satisfied if \( M \) is a diagonal matrix with non-zero entries in the principal diagonal.

Therefore, the correct answer is (C).