Here you will learn what is singular matrix definition with examples and also determinant of singular matrix.
Let’s begin –
Singular Matrix
Definition : A square matrix is a singular matrix if its determinant is zero.
Otherwise, it is a non-singular matrix.
Also Read : How to Find the Determinant of Matrix
Example : Show that the matrix A = \(\begin{bmatrix} 1 & -3 & 4 \\ -5 & 2 & 2 \\ 4 & 1 & -6 \end{bmatrix}\) is singular ?
Solution : The matrix A is singular, if
|A| = 0
\(\implies\) |A| = \(\begin{bmatrix} 1 & -3 & 4 \\ -5 & 2 & 2 \\ 4 & 1 & -6 \end{bmatrix}\)
= 1 \(\begin{vmatrix} 2 & 2 \\ 1 & -6 \end{vmatrix}\) – ( -3) \(\begin{vmatrix} -5 & 2 \\ 4 & -6 \end{vmatrix}\) + 4 \(\begin{vmatrix} -5 & 2 \\ 4 & 1 \end{vmatrix}\)
= 1(-12 – 2) + 3(30 – 8) + 4(-5 – 8)
= -14 + 66 – 52
= 0
\(\implies\) |A| = 0,
Hence, Matrix A is singular.
Example : For what value of x the matrix A = \(\begin{bmatrix} 1 & -2 & 3 \\ 1 & 2 & 1 \\ x & 2 & -3 \end{bmatrix}\) is singular ?
Solution : The matrix A is singular, if
|A| = 0
\(\implies\) \(\begin{vmatrix} 1 & -2 & 3 \\ 1 & 2 & 1 \\ x & 2 & -3 \end{vmatrix}\) = 0
\(\implies\) 1 \(\begin{vmatrix} 2 & 1 \\ 2 & -3 \end{vmatrix}\) + 2 \(\begin{vmatrix} 1 & 1 \\ x & -3 \end{vmatrix}\) + 3 \(\begin{vmatrix} 1 & 2 \\ x & 2 \end{vmatrix}\) = 0
\(\implies\) (-6 – 2) + 2(-3 – x) + 3(2 – 2x) = 0
\(\implies\) -8 – 6 – 2x + 6 – 6x = 0
\(\implies\) -8x – 8 = 0 \(\implies\) x = -1
