Question

can any please explain me how to find out the rank of a matrix
l had my weekend exam tomorrow ​

Answer 1

Answer:

Rank of a matrix is defined as the number of linearly independent rows or columns present in a given matrix.

E.g. Let's consider a 3 × 3 matrix.

$\left[\begin{array}{ccc}1&2&3\\3&6&9\\7&8&9\end{array}\right]$

Here, the first row has the elements [ 1 2 3 ] and the second row has the elements [ 3 6 9 ].

Now let us express these in the form of equation in 3 variables. Hence we get:

⇒ [ 1 2 3 ] = 1x + 2y + 3z   ...(i)

⇒ [ 3 6 9 ] = 3x + 6y + 9z  ...(ii)

Now we can see that Eqn. (ii) can be expressed in terms of (i) by taking 3 in common. That is,

⇒ 3x + 6y + 9z = 3 ( 1x + 2y + 3z )

⇒ 3 × (ii) = (i)

So here, we were able to express one row in terms of the other row. Hence they are dependent on each other. Now if we cannot express two or more rows/columns in terms of the other rows/columns, then it is called independent.

Steps involved in calculating the Rank of A Matrix:

Try to make the elements a₂₁, a₃₁ and a₃₂ of the rows zero using elementary operations on row or column. But you can only use either of them and not both.

Let's find the rank of the above example using this method.

$\left[\begin{array}{ccc}1&2&3\\3&6&9\\7&8&9\end{array}\right]$

Lets first make a₂₂ = 3 to become 0.

So lets do the first elementary operation on row:

Step 1: R₂ → 3R₁ - R₂

So the matrix becomes:

$\left[\begin{array}{ccc}1&2&3\\3-3(1)&6-3(2)&9-3(3)\\7&8&9\end{array}\right] = \left[\begin{array}{ccc}1&2&3\\0&0&0\\7&8&9\end{array}\right]$

Now let's manipulate a₃₁ and a₃₂ to become zero.

Step 2: R₃ → R₃ - 7R₁

The new matrix is:

$\left[\begin{array}{ccc}1&2&3\\0&0&0\\7-7(1)&8-7(2)&9-7(3)\end{array}\right] = \left[\begin{array}{ccc}1&2&3\\0&0&0\\0&-6&-12\end{array}\right]$

Now since we need a₃₂ to become zero without changing a₃₁, lets interchange the R₂ and R₃.

Step 3: R₂ → R₃

The new matrix is:

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

Now we have achieved our conditions of a₂₁, a₃₁ and a₃₂ to be zero. Hence now let's count the number of non-zero rows.

In this case, Row 1 and Row 2 are not having elements sum upto zero. Hence number of independent rows = 2

Therefore the Rank of Matrix is 2.

Answer 2

Answer:

Example :-

Find the Rank of a Matrix :-

$\longrightarrow$ $\left[\begin{array}{ccc}1&2&3\\2&3&4\\3&5&7\end{array}\right]$

Solution :-

First we will convert the given matrix into Echelon form and then find a number of non zero rows.

➸ The order of A is 3 × 3. Hence ρ(A) ≤ 3

$\longrightarrow$ A = $\left[\begin{array}{ccc}1&2&3\\2&3&4\\3&5&7\end{array}\right]$

☯ Convert R₂ ⟶ R₂ - 2R₁ and R₃ ⟶ R₃ - 3R₁

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

☯ Again R₃ ⟶ R₃ - R₂

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

In this number non zero rows is 2.

Henceforth, rank of matrix 2.