Non-Singular Matrix

Non Singular matrix is a square matrix whose determinant is a non-zero value. The non-singular matrix property is to be satisfied to find the inverse of a matrix.

Let us learn more about the non-singular matrix, how to find a non-singular matrix, properties, examples of the non-singular matrix.

A non-singular matrix is a square matrix whose determinant is not equal to zero. The non-singular matrix is an invertible matrix, and its inverse can be computed as it has a determinant value.For a square matrix A = \(\begin{bmatrix}a&b\\c&d\end{bmatrix}\), the condition of it being a non singular matrix is the determinant of this matrix A is a non zero value. |A| =|ad - bc| ≠ 0.

The determinant of a matrix helps to find if the matrix is a singular matrix or a non-singular matrix. The determinant of a non-singular matrix is a non-zero value. The determinant of a matrix can be calculated through row or column operations, or by finding the determinant using the cofactor of the elements of the matrix.

Rules For Row and Column Operations of a Determinant

The following rules are helpful to perform the row and column operations on determinants.

• The value of the determinant remains unchanged if the rows and columns are interchanged.
• The sign of the determinant changes, if any two rows or (two columns) are interchanged.
• If any two rows or columns of a matrix are equal, then the value of the determinant is zero.
• If every element of a particular row or column is multiplied by a constant, then the value of the determinant also gets multiplied by the constant.
• If the elements of a row or a column are expressed as a sum of elements, then the determinant can be expressed as a sum of determinants.
• If the elements of a row or column are added or subtracted with the corresponding multiples of elements of another row or column, then the value of the determinant remains unchanged.

For the simplest square matrix of order 1×1 matrix, which only has only one number, the determinant becomes the number itself. Let's learn how to calculate the determinants for the second order, third order matrices.

Calculating 2D Determinants

For any 2d square matrix or a square matrix of order 2×2, we can use the determinant formula to calculate its determinant:

C = \(\left[\begin{array}{ll}a & b \\c & d\end{array}\right]\)

Its 2D determinant can be calculated as:

|C| = \(\left|\begin{array}{ll}a & b \\c & d\end{array}\right|\)

|C| = (a×d) - (b×c)

Calculating 3D Determinants

For any 3d square matrix or a square matrix of order 3×3, this is the procedure to calculate its determinant.

\(C = \left[\begin{array}{ccc}a_{1} & b_{1} & c_{1} \\a_{2} & b_{2} & c_{2} \\a_{3} & b_{3} & c_{3}\end{array}\right] \)

Here the first row is taken to calculate the determinant of the matrix. The elements \(a_1, b_1, c_1\) are multiplied with their respective cofactors, and the summation of the product of the elements with their respective cofactors, gives the value of the determinant of the square matrix. Alternatively, the elements of any particular row or a column of the matrix can be used to find the determinant of the matrix.

|C| = \(a_{1} \cdot\left|\begin{array}{ll}b_{2} & c_{2} \\b_{3} & c_{3}\end{array}\right|-b_{1} \cdot\left|\begin{array}{cc}a_{2} & c_{2} \\a_{3} & c_{3}\end{array}\right|+c_{1} \cdot\left|\begin{array}{ll}a_{2} & b_{2} \\a_{3} & b_{3}\end{array}\right|\)

|C| = \(a_{1}\left(b_{2} c_{3}-b_{3} c_{2}\right)-b_{1}\left(a_{2} c_{3}-a_{3} c_{2}\right)+c_{1}\left(a_{2} b_{3}-a_{3} b_{2}\right)\)

The following are some of the important properties of a non-singular matrix.

• The determinant of a non-singular matrix is a non-zero value.
• The non-singular matrix is also called an invertible matrix because its determinant can be computed.
• The non-singular matrix is a square matrix because determinants can be calculated only for non-singular matrices.
• The product of two non-singular matrices is a non-singular matrix.
• If A is a non-singular matrix, k is a constant, then kA is also a non-singular matrix.

The following terms are helpful in the process of understanding the concept of a non-singular matrix. Let us briefly learn about each of these terms.

Singular Matrix: A matrix having a determinant value of zero is referred to as a singular matrix. For a singular matrix A, |A| = 0. The inverse of a singular matrix does not exist.

Adjoint of Matrix: The adjoint of a matrix is the transpose of the cofactor element matrix of the given matrix.

Minor: The minor is defined for every element of a matrix. The minor of a particular element is the determinant obtained after eliminating the row and column containing this element. For a matrix A = \(\begin{pmatrix} a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}\), the minor of the element \(a_{11}\) is:

Minor of \(a_{11}\) = \(\left|\begin{matrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{matrix}\right|\)

Cofactor: The cofactor of an element is calculated by multiplying the minor with -1 to the exponent of the sum of the row and column elements in order representation of that element.

Cofactor of \(a_{ij}\) = (-1)^{i + j}× minor of \(a_{ij}\).

Determinant: The determinant of a matrix is the single unique value representation of a matrix. The determinant of the matrix can be calculated with reference to any row or column of the given matrix. The determinant of the matrix is equal to the summation of the product of the elements and its cofactors, of a particular row or column of the matrix.

Inverse of a Matrix: The inverse of a matrix is another matrix, which on multiplication with the given matrix gives the multiplicative identity. For a matrix of order 2 × 2, the general formula for the inverse of matrix is equal to the adjoint of a matrix divided by the determinant of a matrix. A^-1 = \(\dfrac{1}{|A|}\) Adj A

Related Topics

The following topics are helpful for a better understanding of the non-singular matrix.

• Order of Matrix
• Invertible Matrix
• Scalar Matrix
• Unit Matrix
• Symmetric Matrix
• Diagonal Matrix
• Orthogonal Matrix
• Skew Symmetric Matrix
• Hermitian Matrix

What Is called a Non-Singular Matrix?

Non Singular matrix is a square matrix whose determinant is a non-zero value. The non-singular matrix property is to be satisfied to find the inverse of a matrix. For a square matrix A = \(\begin{bmatrix}a&b\\c&d\end{bmatrix}\), the condition of it being a non singular matrix is the determinant of this matrix A is a non zero value. |A| =|ad - bc| ≠ 0.

How Do You Find A Non Singular Matrix?

The non singular matrix can be found by calculating its determinant. A matrix whose determinant is a non zero value, is a non singular matrix.

What Is the Difference Between Singular and Non Singular Matrix?

A singular matrix has a determinant value equal to zero, and a non singular matrix has a determinat whose value is a non zero value. The singular matrix does not have an inverse, and only a non singular matrix has an inverse matrix.

Can a Non Square Matrix Be a Non Singular Matrix?

A non square matrix cannot be a non singular matrix. The determinant can be computed only for a square matrix. Hence only for a square matrix whose determinant is a non zero value, is referred as a non singular matrix.