Reduce the given matrix to reduced rowechelon form and hence determine the rank of each matrix.

Reduce the given matrix to reduced rowechelon form and hence...

Key Concepts

Pivot

A pivot is the first nonzero entry in a row of a matrix in its echelon form, and it is usually scaled to be 1 in reduced row echelon form. Pivots are essential in determining the structure of the matrix, including its rank, and they indicate the positions of the linearly independent rows.

Elementary Row Operations

Elementary row operations are the fundamental manipulations used to simplify matrices. These include swapping two rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another row. These operations are crucial because they do not change the row space of the matrix and thus preserve the solutions of the corresponding linear system.

Reduced Row Echelon Form

Reduced row echelon form (RREF) is a type of matrix that has been simplified using row operations so that each leading entry is 1, all entries in the column containing a leading 1 (pivot) are zero except for the 1 itself, and any rows consisting entirely of zeros are at the bottom of the matrix. This standardized form makes it easier to analyze the properties of the matrix, such as solving linear systems and determining rank.

Rank

The rank of a matrix is the number of linearly independent rows (or columns) it contains. It reflects the dimension of the vector space spanned by the matrix's rows or columns. In the context of RREF, the rank is equal to the number of pivot positions present in the matrix.

Transcript

00:01 Okay, so what we have here is a 2x2 matrix where we're asked to get is reduce raw echelon form.

00:08 So based from our previous lessons, we need to convert this first with its raw echelon form to determine its rank and do further raw operations to convert it to reduce row echelon form.

00:20 So reduced raw etulant form has the same operations with raw echelon form but has different criteria.

00:26 So we say that a matrix is in its role.

00:31 Reduce row echelon form if it satisfy the four requirements.

00:36 First, the first non -zero number in the first row is the number one.

00:42 Second, the second row also starts with the number one, which is further to the right than the leading entry in the first row.

00:49 For every subsequent row, the number one must be further to the right.

00:54 So it should look like this, like the staircase.

00:58 Third, the leading entry in each row must be the only non -zero number in its column, and finally, any non -zero rows are placed at the bottom of the matrix.

01:09 See, we can do this by doing series of elementary row operations, including interchanging one row with another, multiplying one row by a non -zero constant, and replacing one row with one row plus a constant times another row.

01:24 So for this example, what we want to do first is to convert it to its row -etachial form.

01:30 So first, we need to swap r1.

01:32 With r2.

01:36 By doing that, we now have 6, 3, and negative 4.

01:48 Next, we want to cancel the leading coefficient of r2.

01:52 We do this by adding 2 thirds of r1 to r2...

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