Elementary matrix operations

To perform an elementary row operation on a A, an n×m matrix, take the following steps:

• To find E, the elementary row operator, apply the operation to an n×n identity matrix.
• To carry out the elementary row operation, premultiply A by E.

Interchange two rows

Suppose 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 I₃:

100 010 001	⇒	010 100 001
I3		E

Then, to interchange the first and second rows of A, we premultiply A by E (R₁ ↔ R₂):

010 100 001	42 67 24	=	0·4 + 1·6 + 0·20·2 + 1·7 + 0·4 1·4 + 0·6 + 0·21·2 + 0·7 + 0·4 0·4 + 0·6 + 1·20·2 + 0·7 + 1·4	=	67 42 24
E	A				B

Multiply a row by a number

Suppose 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 I₃ by 3:

100 010 001	⇒	100 010 003
I3		E

Then, to multiply each element in the third row of A by 3, we premultiply A by E (3 R₃ → R₃):

100 010 003	42 67 24	=	1·4 + 0·6 + 0·21·2 + 0·7 + 0·4 0·4 + 1·6 + 0·20·2 + 1·7 + 0·4 0·4 + 0·6 + 3·20·2 + 0·7 + 3·4	=	42 67 612
E	A				B

N.B. Divide a row by a number

If we want to divide each element in some row of matrix by number n we must multiply each element of this row by 1n.

Multiply a row and add it to another row

Assume 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 two-step process. First, we multiply each element in the first row of the identity matrix I₂ by 4. Next, we add the result of that multiplication to the second row of I₂ to produce E:

10 01	⇒	10 0 + 4·11 + 4·0	=	10 41
I2				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 R₁ + R₂ → R₂):

10 41	42 67	=	1·4 + 0·61·2 + 0·7 4·4 + 1·64·2 + 1·7	=	42 2215
E	A				B

Use elementary row operations to convert matrix A to the upper triangular matrix

A =		4	2	0
		1	3	2
		-1	3	10

Solution:

R₁ ↔ R₂ (interchange the first and second rows)

	4	2	0		~		1	3	2		~
	1	3	2				4	2	0
	-1	3	10				-1	3	10

-4 R₁ + R₂ → R₂ (multiply 1 row by -4 and add it to 2 row); R₁ + R₃ → R₃ (1 row add to 3 row)

~		1	3	2		~		1	3	2		~
		4 + (-4)·1	2 + (-4)·3	0 + (-4)·2				0	-10	-8
		-1 + 1	3 + 3	10 + 2				0	6	12

R₂ / (-2) → R₂ (divide 2 row by -2); R₃ / 6 → R₃ (divide 3 row by 6)

~		1	3	2		~		1	3	2		~
		0	-10/(-2)	-8/(-2)				0	5	4
		0	6/6	12/6				0	1	2

R₂ ↔ R₃ (interchange the 2 row and 3 row)

~		1	3	2		~
		0	1	2
		0	5	4

-5 R₂ + R₃ → R₃ (multiply 2 row by -5 and add it to 3 row);

~		1	3	2		~		1	3	2
		0	1	2				0	1	2
		0	5 + (-5)·1	4 + (-5)·2				0	0	-6