ALGEBRA AND TRIGONOMETRY By DR. INDU JINDAL Deptt. of Maths PGGCG SEC 42 CHD

Elementary Operations and Rank Of A Matrix

Some Definitions A matrix A is said to be a row (column) equivalent to a matrix B if B can be obtained from A after a finite number of elementary row(column) operations i.e. A∼ B.

 

A matrix in echelon form is said to be in the row reduced echelon form if Distinguished elements are equal to 1,and The column which contains the distinguished element has all other elements equal to zero. Remark : Every matrix A is row equivalent to a matrix in the echelon form or row reduced echelon form.

A matrix obtained from an identity matrix , by subjecting it to an elementary operation is called an elementary matrix. It is written as E-matrix. Theorem: Each elementary row (column) operation on a matrix A has the same effect on A as the pre-multiplication (post-multiplication) of A by the corresponding elementary matrix. Lemma: Every elementary row (column) operation on the product of two matrices can be affected by subjecting the pre-factor (post-factor) of the product of the same row (column) operation.

Types of elementary matrices  

Inverse of a Matrix  

Inverse using elementary operations  

Example 1  

Example 2  

Example 3  

Example 4  

Example 5  

Example 6  

Example 7  

Example 8  

Example 9 Prove that a skew-symmetric matrix of odd order cannot be invertible.

Example 10  

Minor of a matrix  

Example 11 Prove that the inverse of a non-singular symmetric matrix is symmetric.

Rank  

Theorems  

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Example 2  

Example 3  

Example 4  

Example 5  

Example 6  

Rank of matrix using Elementary Operations  

 

Example 1  

Example 2  

Row and Column Vectors  

Linearly dependent  

Linearly independent  

Example 1  

Example 2  

Example 3  

Example 4  

Row-rank and Column-rank  

Example  

Equivalence of matrices Two matrices A and B of same order over a field F are said to be equivalent if there exist non-singular matrices P and Q over F such that B=PAQ Theorem The row rank, the column rank and rank of a matrix are equal.

Example 1  

Example 2  

Example 3  

Example 4  

Example 5  

Example 6  

Example 7  

Example 8 Show that if two matrices A and B over same field are of same type and have same rank, then they are equivalent matrices.