What is Singular Matrix?

Singular Matrix

A singular matrix refers to a matrix whose determinant is zero. Furthermore, such a matrix has no inverse. Students can learn more about the singular matrix here.

Properties of Singular Matrix

The matrices are known to be singular if their determinant is equal to the zero. For example, if we take a matrix x, whose elements of the first column are zero.

Then by the rules and property of determinants, one can say that the determinant, in this case, is zero. Therefore, matrix x is definitely a singular matrix.

A singular matrix is non-convertible in nature. What this means is that its inverse does not exist.
As, an inverse of matrix x = adj(x)/[x], (1)
Where adj(x) is adjoint of x and [x] is the determinant of x.
If, [x] = 0 (singular rmatrix), then the matrix x will not exist according to equation (1).

How to Determine If A Matrix is Singular or Non Singular

Singular matrices are quite unique. Such matrices cannot be multiplied with other matrices to achieve the identity matrix.

Non-singular matrices, on the other hand, are invertible. Furthermore, the non-singular matrices can be used in various calculations in linear algebra. This is because non-singular matrices are invertible.

The first step in plenty of linear algebra problems is the determination of whether a matrix is singular or non-singular.

A matrix can be singular, only if it has a determinant of zero. A matrix with a non-zero determinant certainly means a non-singular matrix.

In case the matrix has an inverse, then the matrix multiplied by its inverse will give you the identity matrix.

Moreover, an identity matrix refers to a square matrix which consists of the same dimensions as the original matrix with the ones on the diagonals and zeroes elsewhere.

Most noteworthy, if an individual is able to find an inverse for a matrix, then it is certainly non-singular.

An individual must verify that the matrix meets all the conditions for the invertible matrix theorem. This verification is important to prove whether the matrix is singular or non-singular.

For an n by n square matrix, the matrix must certainly have a non-zero determinant. Furthermore, the rank of the matrix must equal n. Moreover, the matrix must have linearly independent columns. Also, the matrix should be invertible.

Significance of a Singular Matrix

Singular matrices act as a boundary between matrices whose determinants are positive, and those matrices whose determinants are negative. The sign of the determinant has implications in many fields.

An example can be multiplication by matrices with a positive determinant leads to the preservation of the orientation. On the other hand, multiplication by matrices with a negative determinant leads to the reversal of orientation.

Determinant sign relative to the trace certainly plays a significant role in the qualitative behaviour of non-linear ordinary differential solutions. Singular matrices have determinants which are neither positive nor negative. This makes them quite special.

In real life, the singular matrices would happen with vanishing probability. Furthermore, matrices whose determinants are close to zero certainly take place in real life.

Most noteworthy, their behaviour is quite similar to singular matrices. Hence, the study of singular matrix carries huge importance.

Solved Question For You

Q1 Which of the following statements is not true with regards to the singular matrix?

A. It cannot be multiplied with other matrices to achieve the identity matrix
B. This matrix is invertible
C. It refers to a matrix whose determinant is zero
D. It has no inverse

A1 The correct answer is option B., which is “this matrix is invertible”. This is because it is a non-singular matrix which is invertible. The other three options certainly apply to the singular matrix.