Definition of Determinant of Matrix

Determinant of Matrix Calculator

A determinant of a matrix calculator is a tool used to compute the determinant of a given matrix quickly and accurately....

Definition of Determinant of Matrix

Determinant of a Matrix is defined as the sum of products of the elements of any row or column along with their corresponding co-factors. Determinant is defined only for square matrices....

Determinant of a 1×1 Matrix

Let X = [a] be the matrix of order one, then its determinant is given by det(X) = a....

Determinant of 2×2 Matrix

The determinant of any 2×2 square matrix A = [Tex]\begin{bmatrix}a & b\\c & d\end{bmatrix}_{2\times2}  [/Tex]is calculated using the formula |A| = ad – bc....

Determinant of a 3×3 Matrix

Determinant of a 3×3 Matrix is determined by expressing it in terms of 2nd-order determinants. It can be expanded either along rows(R1, R2 or R3) or column(C1, C2 or C3). Consider a  matrix A of order 3×3...

Determinant of 4×4 Matrix

Determining the determinant of a 4×4 matrix involves more complex methods such as expansion by minors or Gaussian elimination. These techniques require breaking down the matrix into smaller submatrices and recursively finding their determinants. While there isn’t a direct formula like Sarrus’ Rule for 3×3 matrices, the process involves systematic calculations based on the properties of determinants....

Determinant of Identity Matrix

An identity matrix is a square matrix in which all the elements of the main diagonal are ones, and all other elements are zeros. For example, a 3×3 identity matrix looks like this:...

Determinant of Symmetric Matrix

A symmetric matrix is a square matrix that is equal to its transpose. In other words, if A is a symmetric matrix, then A = AT. Symmetric matrices have several interesting properties, one of which is that their determinants remain unchanged under transpose....

Determinant of Skew-Symmetric Matrix

A skew-symmetric (or antisymmetric) matrix is a square matrix whose transpose is equal to its negative. In other words, if A is a skew-symmetric matrix, then A = −AT. Skew-symmetric matrices have interesting properties, one of which is that their determinants have specific values based on the order of the matrix....

Determinant of Inverse Matrix

To understand the determinant of the inverse matrix, let’s first define what the inverse of a matrix is....

Determinant of Orthogonal Matrix

An orthogonal matrix is a square matrix whose rows and columns are orthonormal vectors, meaning that the dot product of any two distinct rows or columns equals zero, and the dot product of each row or column with itself equals one. Mathematically, if A is an orthogonal matrix, then AT⋅A=I, where AT denotes the transpose of A and I represents the identity matrix....

Physical Significance of Determinant

Consider a 2D matrix, each column of this matrix can be considered as a vector on the x-y plane.  So, the determinant between two vectors on a 2d plane gives us the area enclosed between them. If we extend this concept, in 3D the determinant will give us the volume enclosed between two vectors....

Laplace Formula for Determinant

Laplace’s formula, is used to expressed the determinant of a matrix in terms of the minors of the matrix....

Properties of Determinants of Matrix

Various Properties of the Determinants of the square matrix are discussed below:...

Solved Questions on Determinant of Matrix

Question 1: If x, y, and z are different. and A = [Tex]\begin{vmatrix} x & x^{2} & 1 + x^{3} \\ y & y^{2} & 1 + y^{3} \\ z & z^{2} & 1 + z^{3} \end{vmatrix} = 0   [/Tex], then show that 1 + xyz = 0....

FAQs on Determinant of Matrix

What is meant by the determinant formula?...