Question

Find the rank of a matrix of order 3 with each element unity

Answer 1

Answer:

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Answer 2

The rank of a matrix of order 3 with each element unity is 1

Given :

A matrix of order 3 with each element unity

To find :

The rank of a matrix of order 3 with each element unity

Concept :

Let A be a non zero matrix of order m × n. The Rank of A is defined to be the greatest positive integer r such that A has at least one non-zero minor of order r

For a non-zero m × n matrix A

0 < rank of A ≤ min {m, n}

For a non-zero matrix A of order n,

rank of A < , or = n according as A is singular or non-singular

Solution :

Step 1 of 2 :

Write down the given matrix

Here we are given a matrix of order 3 with each element unity

Let A be the matrix

Then the matrix is

$\displaystyle \sf A = \begin{pmatrix} 1 & 1& 1\\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}$

Step 2 of 2 :

Find the rank of the matrix

$\displaystyle \sf A = \begin{pmatrix} 1 & 1& 1\\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}$

We observe that second row and third row are same as first row.

So A has only one linearly independent row

Hence rank of the matrix A is 1

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