Elementary Operations of Matrices

Elementary operations of matrices are fundamental techniques used to simplify matrices and solve systems of linear equations efficiently. These operations include row and column transformations such as swapping rows, multiplying a row by a scalar, and adding a multiple of one row to another. Known as elementary row operations, they are essential tools in linear algebra for matrix reduction, finding inverses, and solving equations using methods like Gauss and Gauss-Jordan elimination. Mastery of these operations is crucial in both academics and real-world applications.

1.0What Are the Elementary Row Operations?

Elementary operations on matrices are operations that allow you to manipulate rows (or columns) of a matrix without changing its solution set (if used to solve linear equations). These operations help transform a matrix into a simpler form — typically row-echelon form or reduced row-echelon form.

2.03 Basic Elementary Operations of Matrix

There are three types of elementary row operations:

1. Row Interchange (Swap) – $R_i\leftrightarrow R_j$

Swap two rows of the matrix.

Example:

$\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\Rightarrow R_1\leftrightarrow R_2\Rightarrow \left[\begin{array}{cc}3& 4\\ 1& 2\end{array}\right]$

2. Row Scaling – $R_i\to k{R}_j$

Multiply a row by a non-zero scalar k.

Example:

$\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\Rightarrow R_2\to 2R_2\Rightarrow \left[\begin{array}{cc}1& 2\\ 6& 8\end{array}\right]$

3. Row Replacement – $R_i\to R_i+k{R}_j$

Add a multiple of one row to another.

Example:

$R_2\to R_2-3R_1$

$\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\Rightarrow \left[\begin{array}{cc}1& 2\\ 0& -2\end{array}\right]$

These operations are reversible and preserve the solution of the system when applied to the augmented matrix of a linear system.

3.0How to Perform Elementary Row Operations Using Matrices

Elementary row operations are typically used in:

• Gauss Elimination: To reduce a matrix to row echelon form.
• Gauss-Jordan Elimination: To reduce a matrix to reduced row echelon form.
• Finding the inverse of a matrix
• Solving linear systems using augmented matrices

Steps:

1. Use row operations to form zeros below the leading 1 (pivot) in each row.
2. Continue until the matrix is in triangular or identity form.
3. Back-substitute if solving a system.

Elementary Row Operations to Transform the Matrix (Example)

Given matrix:

$A=\left[\begin{array}{cc}2& 1\\ 4& 3\end{array}\right]$

We want to transform it to row-echelon form:

Step 1: Make leading 1 in row 1

$R_1\to \frac{1}{2}{R}_1$

$\left[\begin{array}{cc}1& 0.5\\ 4& 3\end{array}\right]$ 

Step 2: Eliminate below pivot using

$\left[\begin{array}{cc}1& 0.5\\ 0& 1\end{array}\right]$

Matrix is now in row-echelon form using elementary row operations.

4.0Applications of Elementary Row Operations

• Solving systems of linear equations
• Finding rank of a matrix
• Matrix inversion
• Checking matrix consistency
• Performing LU decomposition

5.0Solved Examples on Elementary Operations of Matrices

Example 1: If $A=\left[\begin{array}{cc}2& 4\\ 1& 3\end{array}\right]$. Find the inverse of A using elementary row operations.

Solution:

Form an augmented matrix [A | I]:

$\left[\begin{array}{cccc}2& 4& 1& 0\\ 1& 3& 0& 1\end{array}\right]$

Step 1: $R_1\to R_1-2R_2$

$\left[\begin{array}{cccc}0& -2& 1& -2\\ 1& 3& 0& 1\end{array}\right]$

Step 2: Swap $R_1\leftrightarrow R_2$

$\left[\begin{array}{cccc}1& 3& 0& 1\\ 0& -2& 1& -2\end{array}\right]$

Step 3: $R_2\to \frac{R_2}{-2}$

$\left[\begin{array}{cccc}1& 3& 0& 1\\ 0& 1& -\frac{1}{2}& 1\end{array}\right]$

Step 4: $R_1\to R_1-3R_2$

$\left[\begin{array}{cccc}1& 0& \frac{3}{2}& -2\\ 0& 1& -\frac{1}{2}& 1\end{array}\right]$

So,

$\left[\begin{array}{cc}\frac{3}{2}& -2\\ -\frac{1}{2}& 1\end{array}\right]$

Example 2: Reduce the matrix $A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 4& 6\end{array}\right]$ to row echelon form.

Solution:

Apply $R_2\to R_2-2R_1$ :

$\left[\begin{array}{ccc}1& 2& 3\\ 0& 0& 0\end{array}\right]$

This is the row echelon form.

Example 3: If  $A=\left[\begin{array}{ccc}1& 2& -1\\ 2& 5& 0\\ -1& -1& 2\end{array}\right]$. Use elementary operations to find the rank of matrix A.

Solution:
Use row operations to reduce to echelon form:

• $R_2\to R_2-2R_1$
• $R_3\to R_3+R_1$
• Further reduce if possible...

Final row-echelon form (after simplification):$A=\left[\begin{array}{ccc}1& 2& -1\\ 0& 1& 2\\ 0& 0& 1\end{array}\right]$

Since all rows are non-zero, Rank = 3.

Example 3: Use elementary row operations to find the inverse of   $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$

Solution:

Form the augmented matrix [A | I]:

$A=\left[\begin{array}{cccc}1& 2& 1& 0\\ 3& 4& 0& 1\end{array}\right]$

Step 1: $R_2\to R_2-3R_1$

$A=\left[\begin{array}{cccc}1& 2& 1& 0\\ 0& -2& -3& 1\end{array}\right]$

Step 2: $R_2\to \frac{R_2}{-2}$

$A=\left[\begin{array}{cccc}1& 2& 1& 0\\ 0& 1& \frac{3}{2}& -\frac{1}{2}\end{array}\right]$ 

Step 3: $R_1\to R_1-2R_2$

$A=\left[\begin{array}{cccc}1& 0& -2& 1\\ 0& 1& \frac{3}{2}& -\frac{1}{2}\end{array}\right]$

So, $A^{-1}=\left[\begin{array}{cc}-2& 1\\ \frac{3}{2}& -\frac{1}{2}\end{array}\right]$

Example 4: Reduce the matrix to row echelon form$A=\left[\begin{array}{ccc}2& 4& 6\\ 1& 3& 2\\ 3& 5& 8\end{array}\right]$

Solution:

Step 1: $R_1\to \frac{R_1}{2}$

$A=\left[\begin{array}{ccc}1& 2& 3\\ 1& 3& 2\\ 3& 5& 8\end{array}\right]$

Step 2: $R_2\to R_2-R_1,R_3\to R_3-3R_1$

$A=\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& -1\\ 0& -1& -1\end{array}\right]$

Step 3: $R_3\to R_3+R_2$

$A=\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& -1\\ 0& 0& -2\end{array}\right]$

This is the row echelon form.

Example 5: Find the rank using elementary row operations $A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$

Solution:

Step 1: $R_2\to R_2-4R_1,R_3\to R_3-7R_1$

$A=\left[\begin{array}{ccc}1& 2& 3\\ 0& -3& -6\\ 0& -6& -12\end{array}\right]$

Step 2: $R_3\to R_3-2R_2$

$A=\left[\begin{array}{ccc}1& 2& 3\\ 0& -3& -6\\ 0& 0& 0\end{array}\right]$

Only 2 non-zero rows, so Rank = 2.

Example 6: Check if the matrix is invertible using row operations  $A=\left[\begin{array}{cc}2& 4\\ 1& 2\end{array}\right]$

Notice: $Det(A)=2.2-1.4=0$

Or via row operation:

$R_2\to R_2-\frac{1}{2}{R}_1\Rightarrow \left[\begin{array}{cc}2& 4\\ 0& 0\end{array}\right]$

Since the matrix cannot be reduced to identity, it's non-invertible.

Example 7: Use elementary operations to transform $A=\left[\begin{array}{cc}1& 1\\ 2& 3\end{array}\right]$ into the identity matrix.

Solution:

Augment with identity:

$A=\left[\begin{array}{cccc}1& 1& 1& 0\\ 2& 3& 0& 1\end{array}\right]$

Step 1: $R_2\to R_2-2R_1$

$A=\left[\begin{array}{cccc}1& 1& 1& 0\\ 0& 1& -2& 1\end{array}\right]$

Step 2: $R_1\to R_1-R_2$

$A=\left[\begin{array}{cccc}1& 0& 3& -1\\ 0& 1& -2& 1\end{array}\right]$

Thus,

$A^{-1}=\left[\begin{array}{cc}3& -1\\ -2& 1\end{array}\right]$

6.0Practice Questions on Elementary Operations of Matrices

1. Use elementary row operations to find the inverse of: $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$
2. Reduce the matrix to row echelon form: $A=\left[\begin{array}{ccc}1& 3& 2\\ 2& 6& 4\end{array}\right]$
3. Find the rank of the matrix using elementary operations: $A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$
4. Determine whether the following matrix is invertible: $A=\left[\begin{array}{cc}2& 4\\ 1& 2\end{array}\right]$ (Hint: Use row reduction to identity matrix)
5. Transform the matrix $A=\left[\begin{array}{cc}2& 1\\ 4& 3\end{array}\right]$ into an identity matrix using row operations.

7.0Frequently Asked Questions (FAQs)

Q1. How many types of elementary operations exist?

Ans: There are three:

1. Row interchange: $R_i\leftrightarrow R_j$
2. Row scaling: $R_1\to k{R}_1$
3. Row replacement: $R_i\to R_i+k{R}_j$