Properties of a Scalar Matrix

Following are the properties of the scalar matrix

• As transpose of a scalar matrix is equal to the matrix itself, it is a symmetric matrix.
• As the entries above and below the principal diagonal are zero in a scalar matrix, it is both an upper triangular matrix and a lower triangular matrix.
• An identity matrix or a unit matrix is a scalar matrix.
• Any scalar matrix can be obtained when an identity matrix is multiplied by a constant numeric value.
• The determinant of a scalar matrix of any order is equal to the product of the principal diagonal elements.
• The inverse of a scalar matrix is also a scalar matrix whose principal diagonal elements are the reciprocals of the numbers of the original matrix. Remember that the inverse of a scalar matrix exists only if all the principal diagonal elements are not equal to zero.

If A = [Tex]\left[\begin{array}{cc} k & 0\\ 0 & k \end{array}\right][/Tex], then A^-1 = [Tex]\left[\begin{array}{cc} \frac{1}{k} & 0\\ 0 & \frac{1}{k} \end{array}\right][/Tex] (for k ≠ 0).

Articles Related to Scalar Matrix:

• Minors and Cofactors of Determinants
• Adjoint of a Matrix
• Determinant of a Matrix

Scalar matrix is a type of diagonal matrix that has all the elements the same or equal. The elements that are present other than in the diagonal are zero.

In this article, we have covered the definition of scalar matrix, its properties, formula, examples and others in detail.