Rank of Matrix Using Transformation | Normal Form | in Hindi by GP Sir

Summary

In this engaging engineering mathematics lecture, Dr. Gajendra Purohit introduces the concept of the rank of a matrix, a foundational topic in linear algebra. He explains the definition of a matrix, its types, and the importance of rank in solving linear equations. Using multiple examples, Dr. Purohit demonstrates how to determine the rank of matrices through methods like the determinant, minor, and row/column transformations. The lecture emphasizes practical techniques like echelon form and normal form to quickly determine rank, setting the stage for upcoming topics on consistent and inconsistent linear equations.

Takeaways

• 😀 Introduction to the topic of 'Matrices' in Engineering Mathematics, with a focus on understanding the rank of a matrix.
• 😀 A matrix is defined by the number of rows (m) and columns (n), and the order of a matrix is denoted as mxn.
• 😀 Square matrices have an equal number of rows and columns (e.g., 2x2, 3x3, 4x4 matrices).
• 😀 The rank of a matrix refers to the number of different rows in the matrix, which defines its rank value (e.g., 3x3 matrix with all distinct rows has rank 3).
• 😀 The rank of a matrix can be reduced when rows are identical or dependent. If two rows are identical, the rank is reduced to 2, and so on.
• 😀 Minors (sub-matrices) are used to calculate the determinant. If the determinant of the minors is non-zero, it helps in determining the rank of the matrix.
• 😀 An 'Echelon form' matrix can be used to determine the rank of a matrix. If the last row remains non-zero, the rank is equal to the number of non-zero rows.
• 😀 In some cases, the rank is found using transformations (row and column transformations), with an emphasis on simplifying the matrix into a form that makes the rank clearer.
• 😀 Reducing a matrix to its 'normal form' involves applying transformations to create a unit matrix of the same rank (e.g., a 2x2 unit matrix if the rank is 2).
• 😀 The process of column transformation is also explained as part of matrix rank determination, particularly when reducing to normal form for clarity.
• 😀 In conclusion, understanding the rank of a matrix is fundamental to topics like 'Consistent and Inconsistent Linear Equations,' which will be explored in the next lesson.

Q & A

What is the rank of a matrix?

The rank of a matrix is the number of distinct, non-zero rows in the matrix. It can also be defined as the largest number of linearly independent rows or columns in the matrix.

How is the rank of a square matrix determined?

For a square matrix, the rank is determined by the number of distinct rows. If all rows are distinct, the rank is equal to the size of the matrix (e.g., 3 for a 3x3 matrix). If some rows are identical, the rank is less than the size of the matrix.

What is the significance of minors in determining the rank of a matrix?

Minors are sub-matrices obtained by removing one or more rows and columns from the original matrix. If the determinant of a minor is non-zero, it indicates that the rank of the matrix is at least the order of that minor.

What are the two common methods for finding the rank of a matrix?

The two common methods for finding the rank of a matrix are the Determinant Method (Minor Method), where the rank is determined using the determinants of minors, and Row/Column Transformation (Echelon Form), where the matrix is simplified to identify its rank.

What is Echelon form in matrix rank determination?

Echelon form is a simplified version of a matrix where all non-zero rows are above any rows of zeros, and the leading coefficient of each row is to the right of the leading coefficient of the row above it. This form helps in determining the rank of the matrix.

What happens if you apply transformations to a matrix and end up with a row of zeros?

If applying transformations results in a row of zeros, the rank of the matrix is reduced by the number of rows of zeros. The remaining non-zero rows will determine the rank.

Can the rank of a matrix ever be greater than its order (e.g., rank > 3 for a 3x3 matrix)?

No, the rank of a matrix cannot exceed its order. For a square matrix of size mxn, the maximum possible rank is the smaller of m or n.

What is the normal form of a matrix and how is it related to rank?

The normal form of a matrix is a simplified form where the matrix is reduced to a unit matrix of the same rank. For example, if the rank of a matrix is 2, it will be reduced to a 2x2 unit matrix with all other elements as zero.

What do you do if you're asked to find the rank of a matrix by reducing it to normal form?

If asked to find the rank by reducing it to normal form, first determine the rank using methods like row/column transformations. Then, reduce the matrix to a unit matrix of the determined rank, making all other elements zero.

How can you use column transformation to find the rank of a matrix?

Column transformation is used to simplify the matrix by applying operations on the columns, such as interchanging columns or scaling them. This method helps in reducing the matrix to a simpler form, making it easier to identify its rank.