Made easy test

Question

Consider the rank of matrix $'A'$ of size $(m \times n)$ is $"m-1"$. Then, which of the following is true?

• A. $AA^T$ will be invertible.
• B. $A$ have $"m-1"$ linearly independent rows and $"m-1"$ linearly independent column.
• C. $A$ will have $"m"$ linearly independent rows and $"n"$ linearly independent columns.
• D. $A$ will have $"m-1"$ linearly independent rows and $"n-1"$ independent columns.

Comments

A matrix A is given. Rank of A is m-1. Then it is asking which of the following options are correct.

Do you have doubt regarding why option B is correct.

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Yes

Answer 1

Here it is given rank of matrix $m-1$

That means there are $m-1$ non zero rows in the matrix 

As, if the definition of linearly independent is the determinant of the matrix must be non-zero, otherwise in non linearly independent matrix determinant of the matrix must be 0. 

Now, as the matrix has rank $m-1$ , that means matrix is a square matrix

Because, without square matrix we cannot find determinant

From here we can say, there must be $m-1$ column which are non-zero.

So, option B) is correct

for more information here

Comments

Thanks for the nice explanation.

Answer 2

these basics of rank is useful to do such theoritical problems.

Answer 3

Rank of matrix A=The number of linearly independent rows of A=the number of linearly independent columns of A

As in question rank of matrix is given m-1 so it will have m-1 linearly independent rows and m-1 linearly independent columns

Therefore option B is correct