Determinant of a matrix changes its sign if we interchange any two rows or columns present in a matrix. We can prove this property by taking an example. We take matrix A and we calculate its determinant (|A|).
Properties of Adjoint of Matrices
This post is dedicated to some important properties regarding adjoint of matrix. If, you want to go through their proves then click particular property. Note that these properties are only valid for square matrices as adjoint is only valid for square matrices.
where, A is a square matrix, I is an identity matrix of same order as of A and
represents determinant of matrix A.
determinant of adjoint A is equal to determinant of A power n-1 where A is invertible n x n square matrix.
{A is n x n invertible square matrix}
You can also take examples to verify these properties.
Multiplying Scalar with a Matrix
To multiply any scalar with a matrix, we simply multiply every element present in the matrix with that scalar. For example, if we are given with any matrix A and we want to calculate 2A then we simply multiply each element of matrix with 2 to get the resultant matrix. Lets take an example of matrix A =
Suppose, we want to calculate 2A. Just Multiply each element of A with 2 to get 2A.
2A =
Similarly, we can calculate kA by multiplying each element of matrix A with k.
The order of matrix kA is same as that of A.
There are some useful properties regarding multiplication of scalars with matrices:
provided that A and B have same order.
{Click on property to see its proof}
{Click on property to see its proof}
How to Add or subtract Two Matrices?
How to Add or subtract two Matrices?
Two matrices can be added or subtracted only if they have same order. Otherwise, they are not eligible for addition. For example, we have two matrices
and
, they are eligible to be added or subtracted because they have same order (2 x 3).
Starting with Matrices (Order of Matrix)
What is a matrix?
A matrix is a rectangular array of elements or numbers. A matrix is usually enclosed in square or round brackets.
For example, Matrix of dimensions or order of 3 x 3 can be represented like:
A=