Singular Matrix

We determine whether a matrix is a singular matrix or a non-singular matrix depending on its determinant. The determinant of a matrix 'A' is denoted by 'det A' or '|A|'. If the determinant of a matrix is 0, then it is said to be a singular matrix. Why do we need to have a specific name for the matrices with determinant 0? Let us see.

Let us learn more about the singular matrix along with its definition, formula, properties, and examples.

A singular matrix is a square matrix if its determinant is 0. i.e., a square matrix A is singular if and only if det A = 0. We know that the inverse of a matrix A is found using the formula A^-1 = (adj A) / (det A). Here det A (the determinant of A) is in the denominator. We are aware that a fraction is NOT defined if its denominator is 0. Hence A^-1 is NOT defined when det A = 0. i.e., the inverse of a singular matrix is NOT defined. i.e., there does not exist any matrix B such that AB = BA = I (where I is the identity matrix).

From the above explanation, a square matrix 'A' is said to be singular if

• det A = 0 (which is also written as |A| = 0) (or)
• A^-1 is NOT defined (i.e., A is non-invertible).

To determine/identify whether a given matrix is singular we need to check for two conditions:

• check whether A is a square matrix.
• check whether det A = 0.

Here are few examples to find whether a given matrix is singular.

• A = \(\left[\begin{array}{rr}3 & 6 \\ \\ 2 & 4 \end{array}\right]\) is a singular matrix because
  it is a square matrix (of order 2 × 2) and
  det A (or) |A| = 3 × 4 - 6 × 2 = 12 - 12 = 0.
• A = \(\left[\begin{array}{rr}1 & 2 & 2 \\ 1 & 2 & 2\\ 3 & 2&-1 \end{array}\right]\) is a singular matrix because
  it is a square matrix (of order 3 × 3) and
  as det A (or) |A| = 0 (as the first two rows are equal).

Here are some singular matrix properties based upon its definition.

• Every singular matrix is a square matrix.
• The determinant of a singular matrix is 0.
• The inverse of a singular matrix is NOT defined and hence it is non-invertible.
• By properties of determinants, in a matrix,
  * if any two rows or any two columns are identical, then its determinant is 0 and hence it is a singular matrix.
  * if all the elements of a row or column are zeros, then its determinant is 0 and hence it is a singular matrix.
  * if one of the rows (columns) is a scalar multiple of the other row (column) then the determinant is 0 and hence it is a singular matrix.
• A null matrix of any order is a singular matrix.
• The rank of a singular matrix is definitely less than the order of the matrix. For example, the rank of a 3x3 matrix is less than 3.
• All rows and columns of a singular matrix are NOT linearly independent.

A non-singular matrix, as its name suggests, is a matrix that is NOT singular. Thus, the determinant of a non-singular matrix is a nonzero number. i.e., a square matrix 'A' is said to be a non singular matrix if and only if det A ≠ 0. Then it is obvious that A^-1 is defined. i.e., a non-singular matrix always has a multiplicative inverse. Thus, we can summarize the differences between the singular matrix and non-singular matrix as follows:

There is one important theorem on singular matrix that can actually be used to generate a singular matrix and the theorem says: "The product of two matrices A = [A]_{n × k} and B = [B]_{k × n} (where n > k) is a matrix AB of order n × n and is always singular". By this theorem:

• The product AB of two matrices A of order n × 1 and B of order 1 × n is singular always.
• The product AB of two matrices A of order n × 2 and B of order 2 × n is also singular, etc.

Using this theorem, one can generate a singular matrix by multiplying two randomly generated matrices of orders n × k and k × n where n > k.

☛ Related Topics:

• Diagonal Matrix
• Orthogonal Matrix
• Inverse of 3x3 Matrix
• Symmetric Matrix

What Does a Singular Matrix Mean?

A singular matrix means a square matrix whose determinant is 0 (or) it is a matrix that does NOT have a multiplicative inverse.

How do You Know if a Matrix is Singular Matrix?

We can say that a matrix 'A' is singular if one of the following conditions is satisfied.

• If determinant of A = 0 (or)
• If A is non-invertible.

What is Singular Matrix Formula?

We know that a singular matrix determinant is 0. Thus, the formula for the singular matrix is "A is singular if and only if det(A) = 0".

Why is It Called a Singular Matrix?

The word "singular" means "exceptional" (or) "remarkable". A singular matrix is specifically used to determine whether a matrix has an inverse, rank of a matrix, uniqueness of the solution of a system of equations, etc. It is also used for various purposes in linear algebra and hence the name.

What is a Singular Matrix 3x3?

The determinant of a singular matrix is 0. An example of a 3x3 singular matrix is \(\left[\begin{array}{rr}2 & 1 & -1 \\ 1 & 0 & 1\\ 2 & 1&-1 \end{array}\right]\) is singular as its determinant is zero (as its first and third rows are equal).

What Makes a Matrix A a Singular Matrix?

If the determinant of A is 0 then A is singular. If there is no matrix B such that AB = BA = I, then it means that A has no inverse and in this case also, A is said to be singular.

What is the Rank of a Singular Matrix?

If A is a singular matrix of order n, then it means that det A = 0. Then the rank of the matrix is definitely less than the order of the matrix. i.e., rank(A) < n.

What are Singular and Non Singular Matrices?

A singular matrix is a matrix whose determinant is 0 and hence it has no inverse. On the other hand, a non-singular matrix is a matrix whose determinant is NOT 0 and hence it has an inverse.

What is Singular Matrix Determinant?

A singular matrix has no inverse. We know that the inverse of a matrix A is (adj A)/(det A) and it does NOT exist when det A = 0. Therefore, the determinant of a singular matrix is 0.