There are many important properties regarding transpose of Matrices. You need them often to solve different types of problems.
Properties of Transpose of Matrices:
(5)
Click on the property to see its proof.
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We are given matrix A, how can we find its transpose. To find transpose of matrix, it is necessary to understand definition of transpose of matrix.
If
then its transpose which is written as
When we want to calculate value of f(a) when a is any constant, we simply replace value of x by a in the given function f(x). We follow the same procedure to calculate value of f(A) where A is any matrix. We replace value of x by A and do the required calculations.
What are Nilpotent, Involutory and Idempotent matrices?
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}