Singular Matrix – Definition, Examples and Determinant

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

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