How to Find the Rank of a Matrix Using Normal Form

TL;DR

The rank of a matrix equals the number of non-zero (different) rows left after row transformations reduce it to echelon form. For a 3x3 matrix, all rows different gives rank 3, two identical rows gives rank 2, and all rows identical gives rank 1. When a question asks for normal form, apply column transformations until an identity matrix of that rank appears with all other elements zero.

Transcript

Hello students, I'm Dr. Gajendra Purohit and today I'm starting the classes of Engineering mathematics I welcome you all to the classes So the first topic in engineering mathematics is 'matrices' Its a part of linear algebra The topics of matrices that come in B.Sc and engineering mathematics are 'Rank of a matrix', 'Inconsistent and consistent lin... Read More

Key Insights

• The rank of a matrix is the number of different rows it contains. If a 4x4 matrix has four different rows the rank is 4, two identical rows gives rank 3, three identical rows gives rank 2, and all four rows identical gives rank 1.
• The formal definition of rank is the largest order of any non-vanishing minor of the matrix, denoted R(A). Some sub-matrix determinant must be non-zero, while the determinant of every sub-matrix of the next higher order must be zero.
• Minors are the sub-matrices a matrix can be divided into. In the worked example all the 2x2 minors have non-zero determinants but the 3x3 determinant is zero, so the rank of that matrix is 2.
• The determinant or minor method for finding rank has a serious limitation: it cannot determine the rank of anything other than a square matrix. For non-square matrices such as 3x4 matrices, row and column transformation must be used instead.
• Echelon form is reached by applying row transformations until the lower rows become zero. If the last row is still non-zero the rank is 3; if the transformation drives every element of the last row to zero the rank drops to 2.
• Normal form means creating a unit matrix of the same size as the rank, with all remaining elements set to zero. A matrix of rank 2 must be reduced to a 2x2 unit matrix, and a matrix of rank 3 to a 3x3 unit matrix.
• Column transformation is the tool used for reducing a matrix to normal form, while row transformation is what produces echelon form. The rank found from the row work tells you the size of the unit matrix you are aiming to build with the columns.
• Interchanging columns is a practical speed trick when an awkward element such as 6 sits in the first corner position. Swapping it out avoids difficult calculations and prevents a zero corner element, making the transformation process faster.
• Knowing the target before starting is the key to normal form questions. Even if the rank is already obvious by inspection, the answer should not be written directly; the matrix must actually be reduced, and a clear objective makes the reduction achievable.
• Rank is the foundation for the next topic in the series, consistent and inconsistent linear equations, which depends entirely on the concept of rank.

Questions & Answers

Q: What is the rank of a matrix?

The rank of a matrix is defined as the largest order of any non-vanishing minor of the matrix, denoted R(A). In practical terms it is the number of different rows in the matrix. For a 3x3 matrix, if all three rows are different the rank is 3; if two rows are the same the rank is 2, because both identical rows are counted as one; and if all three rows are the same the rank is 1. The same logic scales to a 4x4 matrix: four different rows gives rank 4, two same rows gives rank 3, three same rows gives rank 2, and all rows the same gives rank 1.

Q: How to find the rank of a matrix using row transformation?

Apply row transformations to the matrix until it is reduced to echelon form, then count the rows that are still non-zero. In the first worked example the last line of the matrix remains non-zero after transformation, so the rank of that matrix is 3. Had the transformation driven all the elements of the last line to zero, the rank would have been 2 instead. In a later 4x4 example, two of the four lines became zero, leaving a rank of 2. This method works on any matrix, not only square ones, which is why it is preferred over the minor method.

Q: What is normal form of a matrix and how do you reduce a matrix to it?

Normal form means making a unit matrix of the same size as the rank of the matrix, with all the other elements reduced to zero. If the rank of the matrix is 2, you must reach a 2x2 unit matrix with the rest of the elements as zero. If the rank is 3, you must reach a 3x3 unit matrix with the rest as zero. The reduction is carried out using column transformation, in contrast to echelon form, which is produced by row transformation. Determining the rank first tells you exactly what target shape you are working towards.

Q: Why does the determinant or minor method fail for non-square matrices?

The determinant method, also called the minor method, finds rank by checking the determinants of the sub-matrices, or minors, of the original matrix. Because determinants only exist for square matrices, this approach cannot help in determining the rank of matrices other than square ones. A 3x4 matrix, for instance, cannot be handled this way. In such cases row and column transformation is used instead, reducing the matrix to echelon form and counting the non-zero rows to get the rank.

Q: What are minors of a matrix?

A matrix can be divided into many sub-matrices, and these sub-matrices are known as minors. In the example used to explain the definition of rank, four minors labelled m1, m2, m3 and m4 are all 2x2 sub-matrices, and each of them has a non-zero determinant. The 3x3 determinant of the full matrix, however, is zero. This matches the definition of rank exactly: the largest order with a non-zero minor is 2 while the next higher order gives zero, so the rank of that matrix is 2. This is confirmed by the shortcut, since two of its rows are the same.

Q: When should you interchange columns while finding the rank of a matrix?

Interchange columns when an inconvenient element sits in the first corner position and would complicate the calculation. In both 4x4 examples in the lesson the first element was 6, which can create a problem during transformation, so two columns were interchanged before proceeding. The same swap is also used to avoid having a zero as the corner element. This is purely a practical speed measure: it keeps the arithmetic simple and makes the transformation process faster without changing the rank of the matrix.

Q: What is the difference between echelon form and normal form?

Echelon form is reached by applying row transformations until the matrix has zeros below the leading elements, and the rank is read off by counting the rows that are still non-zero. Normal form is reached by applying column transformations to produce a unit matrix whose size equals the rank, with every remaining element set to zero. Both give the same rank, but exams sometimes specifically ask you to find the rank by reducing to normal form. In that case, even when the rank is already obvious by inspection, the matrix must actually be reduced rather than the answer written down directly.

Q: What topics does this engineering mathematics matrices chapter cover?

The matrices chapter that appears in B.Sc and engineering mathematics covers rank of a matrix, inconsistent and consistent linear equations, eigen values and eigen vectors, and the Cayley Hamilton theorem. Matrices form part of linear algebra and are the first topic in engineering mathematics. This lesson covers rank of a matrix, starting with basics such as the m x n order notation and the definition of a square matrix as one where the number of rows equals the number of columns. The next topic in the series is consistent and inconsistent linear equations, which depends entirely on the concept of rank.

Summary & Key Takeaways

• Matrices are the first topic of engineering mathematics and part of linear algebra, with rank of a matrix, consistent and inconsistent linear equations, eigen values and eigen vectors, and the Cayley Hamilton theorem forming the chapter. A matrix is written with m rows and n columns, where m x n is its order, and a matrix with equal rows and columns is a square matrix.

• Rank is the number of different rows in a matrix. A 3x3 matrix with all rows different has rank 3, with two rows the same has rank 2, and with all rows the same has rank 1. Formally, rank is the largest order of a non-vanishing minor: some sub-matrix determinant is non-zero while every determinant of the next higher order is zero.

• Two main approaches are shown. The determinant or minor method uses sub-matrices but only works for square matrices. Row and column transformation reduces the matrix to echelon form, and the count of remaining non-zero rows gives the rank. Four worked examples cover a 3x4 matrix, the same matrix reduced to normal form, and two 4x4 matrices.

• Normal form means building a unit matrix whose size equals the rank, with all other elements zero. A rank 2 matrix reduces to a 2x2 unit matrix, a rank 3 matrix to a 3x3 unit matrix. Column transformations are used for this step. In the 4x4 examples, columns are interchanged first so that a 6 does not sit in the corner position and slow the calculation.