Symmetric and Skew Symmetric Matrices

The video tutorial explains symmetric and skew symmetric matrices, defining each and providing examples. It discusses theorems related to these matrices and demonstrates how any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix. An example is provided to illustrate this decomposition process.

What is a symmetric matrix?

• A matrix with non-zero diagonal elements

• A matrix that is equal to its transpose

• A matrix that is equal to the negative of its transpose

• A matrix with all zero elements

Which of the following is true for a skew symmetric matrix?

• All diagonal elements are non-zero

• It is equal to its transpose

• It is equal to the negative of its transpose

• It has no zero elements

What is the condition for a matrix to be skew symmetric?

• It must be equal to the negative of its transpose

• It must have all non-zero elements

• It must be equal to its transpose

• It must have all zero diagonal elements

What is the result of the product of a matrix and its transpose?

• Always an identity matrix

• Always a skew symmetric matrix

• Always a symmetric matrix

• Always a zero matrix

How can any square matrix be expressed?

• As the difference of two skew symmetric matrices

• As the product of two symmetric matrices

• As the sum of two identity matrices

• As the sum of a symmetric and a skew symmetric matrix

What is the symmetric part of a matrix decomposition?

• The part with non-zero diagonal elements

• The part that is equal to the negative of its transpose

• The part that is equal to its transpose

• The part with all zero elements

In the decomposition of a matrix, what is the skew symmetric part?

• The part with all zero elements

• The part that is equal to the negative of its transpose

• The part that is equal to its transpose

• The part with non-zero diagonal elements