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  4. The matrix | 5 10 3 -2 -4 6 -1 -2 b | is a singular matrix, if is equal to:

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Q. The matrix $ \begin{vmatrix} 5 & 10 & 3 \\ -2 & -4 & 6 \\ -1 & -2 & b \\ \end{vmatrix} $ is a singular matrix, if is equal to:

Jharkhand CECEJharkhand CECE 2005

Solution:

If any square matrix is singular, then the value of determinant is zero.
Let $ A= \begin{vmatrix} 5 & 10 & 3 \\ 2 & -4 & 6 \\ -1 & -2 & b \\ \end{vmatrix} $
Since, $ A $ is singular.
$ \therefore \begin{vmatrix} 5 & 10 & 3 \\ -2 & -4 & 6 \\ -1 & -2 & b \\ \end{vmatrix} =0 $
$ \Rightarrow 5(-4b+12)-10(-2b+6)+3(4-4)=0 $
$ \Rightarrow -20b+60+20b-60+0=0 $
$ \Rightarrow 0=0 $
$ \therefore $ The given matrix is singular for any value of $ b $ .
For a non-singular matrix, the value of determinant is non-zero.