Consider the matrix $M=\left[\begin{array}{ccc}2& 1& 1\\ 1& 3& 0\\ -1& a& b\end{array}\right]$. Which of the following options is/ are TRUE if $\det \left(M\right)\ne 0$ ?

The correct options are a = 1/2 and b = 1/2 and a = 1/2 and b = -3.

To determine which options are true when the determinant of the matrix M is non-zero, let us first calculate the determinant of matrix M. The matrix is given by:

$M=\left[\begin{array}{ccc}2& 1& 1\\ 1& 3& 0\\ -1& a& b\end{array}\right]$

We expand the determinant along the first row:

$\det \left(M\right)=2·\det \left[\begin{array}{cc}3& 0\\ a& b\end{array}\right]-1·\det \left[\begin{array}{cc}1& 0\\ -1& b\end{array}\right]+1·\det \left[\begin{array}{cc}1& 3\\ -1& a\end{array}\right]$

Now, we compute the 2 × 2 determinants:

Simplifying the terms, we get:

$\det \left(M\right)=6b-b+a+3$

$\det \left(M\right)=a+5b+3$

We are given that the determinant of matrix M is non-zero, which means:

$a+5b+3\ne 0$

Now, let us evaluate each of the given options to see if they satisfy this condition:

1. Option: a = -1/2 and b = -1/2
Substitute these values into the determinant expression: