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In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation (or column operation). Left multiplication (premultiplication) by an elementary matrix represents elementary row operations, while right multiplication (postmultiplication) represents elementary column operations. Elementary matrix operations play an important role in many matrix algebra applications, such as finding the inverse of a matrix, in Gaussian elimination to reduce a matrix to row echelon form and solving simultaneous linear equations.
Elementary operations:Elementary operations notation:Compact notation to describe elementary operations:
Elementary operatorsEach type of elementary operation may be performed by matrix multiplication, using square matrices called elementary operators.Elementary row operationsTo perform an elementary row operation on a A, an n×m matrix, take the following steps:
Interchange two rowsSuppose we want to interchange the first and second rows of A, a 3×2 matrix. To create the elementary row operator E, we interchange the first and second rows of the identity matrix I_{3}:
Then, to interchange the first and second rows of A, we premultiply A by E (R_{1} ↔ R_{2}):
Multiply a row by a numberSuppose we want to multiply each element in the third row of Matrix A by 3. Assume A is a 3×2 matrix. To create the elementary row operator E, we multiply each element in the third row of the identity matrix I_{3} by 3:
Then, to multiply each element in the third row of A by 3, we premultiply A by E (3 R_{3} → R_{3}):
N.B. Divide a row by a numberIf we want to divide each element in some row of matrix by number n we must multiply each element of this row by Multiply a row and add it to another rowAssume A is a 3×2 matrix. Suppose we want to multiply each element in the first row of A by 4; and we want to add that result to the second row of A. For this operation, creating the elementary row operator is a twostep process. First, we multiply each element in the first row of the identity matrix I_{2} by 4. Next, we add the result of that multiplication to the second row of I_{2} to produce E:
Then, to multiply each element in the first row of A by 4 and add that result to the second row, we premultiply A by E ( 4 R_{1} + R_{2} → R_{2}):
Elementary column operationsTo perform an elementary column operation on a A, an n×m matrix, take the following steps:
Examples of elementary matrix operationsExample 1. Use elementary row operations to convert matrix A to the upper triangular matrix
Solution: R_{1} ↔ R_{2} (interchange the first and second rows)
4 R_{1} + R_{2} → R_{2} (multiply 1 row by 4 and add it to 2 row); R_{1} + R_{3} → R_{3} (1 row add to 3 row)
R_{2} / (2) → R_{2} (divide 2 row by 2); R_{3} / 6 → R_{3} (divide 3 row by 6)
R_{2} ↔ R_{3} (interchange the 2 row and 3 row)
5 R_{2} + R_{3} → R_{3} (multiply 2 row by 5 and add it to 3 row);
MatrixMatrix Definition. Main informationSystem of linear equations  matrix formTypes of matricesMatrix scalar multiplicationAddition and subtraction of matricesMatrix multiplicationTranspose matrixElementary matrix operationsDeterminant of a matrixMinors and cofactors of a matrixInverse matrixLinearly dependent and independent rowsRank of a matrix Add the comment 