Question

Are all matrices Diagonalizable?

Answer 1

Hint: In this question, we have to check whether all matrices are diagonalizable or not. So, a square matrix \[A\] is diagonalizable if it is similar to a diagonal matrix that is if there exists an invertible matrix \[P\] such that \[{P^{ - 1}}AP\] is a diagonal matrix.

Complete step-by-step solution: Matrices are the form of rows and columns that are arranged in a rectangular form. There are a lot of operations that can be performed in the matrices. For example, addition, subtraction, and multiplication. There are different types of matrices that are present. For example,
1) Row matrices.
2) Column matrices.
3) Singular matrices.
4) Symmetric and non-symmetric matrices.
Diagonalization matrices are square matrices that are having \[n \times n\] the form and it is written as,
\[A = PD{P^{ - 1}}\]
Or equivalently say that
\[D = {P^{ - 1}}AP\]
Here, P is a singular matrix, D is the diagonal matrix and \[{P^{ - 1}}\] is the inverse singular matrix.
Now, here we can see that the result is coming in the form of \[n \times n\] matrix which means that it is in the form of square matrices because the row and column are equal but as we can see that the matrices are of different types as well, for example, row-matrices which is the form of \[m \times n\] where \[m\] is the row and \[n\] is the column.

Hence, all matrices are not diagonalizable.

Note : The matrices are square as well as non-square matrices. There are some main properties that are followed by the matrices. The properties are commutative, associative, identity and inverse.