Solving Linear Equations - Matrix Algebra, CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

Mathematics for IIT JAM, CSIR NET, UGC NET

Mathematics : Solving Linear Equations - Matrix Algebra, CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

The document Solving Linear Equations - Matrix Algebra, CSIR-NET Mathematical Sciences Mathematics Notes | EduRev is a part of the Mathematics Course Mathematics for IIT JAM, CSIR NET, UGC NET.
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The matrix method of solving systems of linear equations is just the elimination method in disguise. By using matrices, the notation becomes a little easier.

Suppose you have a system of linear equations such as:

Solving Linear Equations - Matrix Algebra, CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

The first step is to convert this into a matrix. Make sure all equations are in standard form Solving Linear Equations - Matrix Algebra, CSIR-NET Mathematical Sciences Mathematics Notes | EduRev and use the coefficients of each equation to form each row of the matrix. It may help you to separate the right column with a dotted line.

Solving Linear Equations - Matrix Algebra, CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

Next, we use the matrix row operations to change the 2×2 matrix on the left side to the identity matrix . First, we want to get a zero in Row 1 , Column 2 . So, add 4 times Row 2 to Row 1 .

Solving Linear Equations - Matrix Algebra, CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

Next we want a 1 in the top left corner.

Solving Linear Equations - Matrix Algebra, CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

Now we want a zero in the bottom left corner.

Solving Linear Equations - Matrix Algebra, CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

Finally, we want a 1 in Row 2 , Column 2 .

Solving Linear Equations - Matrix Algebra, CSIR-NET Mathematical Sciences Mathematics Notes | EduRev

Now that we have the 2×2 identity matrix on the left, we can read off the solutions from the right column:

x = 3

y = -1

The same method can be used for nn linear equations in n unknowns; in this case you would create an n×(n−1) matrix, and use the matrix row operations to get the identity n×n matrix on the left side.

Important Note: If the equations represented by your original matrix represent parallel lines, you will not be able to get the identity matrix using the row operations. In this case, the solution either does not exist or there are infinitely many solutions to the system.

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