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Matrix Rank and Nullity Concepts

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Questions and Answers

What does the rank of a matrix represent?

Dimension of the image of the matrix (correct)

Trace of the matrix

Dimension of the kernel of the matrix

Determinant of the matrix

Which property does the nullity of a matrix refer to?

Determinant of the matrix

Trace of the matrix

Dimension of the kernel of the matrix (correct)

Dimension of the image of the matrix

In the context of matrices, what does 'dim(im A)' stand for?

Dimension of the kernel of the matrix

Determinant of the matrix

Dimension of the image of the matrix (correct)

Trace of the matrix

How do you determine the nullity of an n × m matrix A?

By counting the number of free variables (A) Signup and view all the answers

What does the rank of an n × m matrix A represent?

The dimension of the image of A (B) Signup and view all the answers

In the context of matrices, what is the relationship between the nullity and the rank?

The rank minus the nullity is equal to n (A) Signup and view all the answers

If A and B are square matrices of size 3×3 and AB has a kernel of dimension 2, what can be said about the kernel of A?

The kernel of A must be of dimension 1. (C) Signup and view all the answers

What is a possible matrix B that, when multiplied by I (the identity matrix), would result in a product with a kernel of dimension 2?

[0 1 0] (C) Signup and view all the answers

If matrix B is square and contains non-zero elements, what property must matrix A have to ensure that the product AB has a kernel of dimension greater than 0?

Matrix A must have linearly dependent rows/columns. (C) Signup and view all the answers

Matrix Rank and Nullity Concepts

Choose a study mode

Podcast

Questions and Answers

What does the rank of a matrix represent?

Which property does the nullity of a matrix refer to?

In the context of matrices, what does 'dim(im A)' stand for?

How do you determine the nullity of an n × m matrix A?

What does the rank of an n × m matrix A represent?

In the context of matrices, what is the relationship between the nullity and the rank?

If A and B are square matrices of size 3×3 and AB has a kernel of dimension 2, what can be said about the kernel of A?

What is a possible matrix B that, when multiplied by I (the identity matrix), would result in a product with a kernel of dimension 2?

If matrix B is square and contains non-zero elements, what property must matrix A have to ensure that the product AB has a kernel of dimension greater than 0?

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