# Linear Algebra : The Identity Matrix and Diagonal Matrices

## Example Questions

### Example Question #1 : Operations And Properties

Which of the following matrices is a scalar multiple of the identity matrix?

Explanation:

The x identity matrix is

For this problem we see that

And so

is a scalar multiple of the identity matrix.

### Example Question #1 : The Identity Matrix And Diagonal Matrices

Which of the following is true concerning diagonal matrices?

The trace of any diagonal matrix is equal to its determinant.

The product of two diagonal matrices (in either order) is always another diagonal matrix.

The determinant of any diagonal matrix is .

The zero matrix (of any size) is not a diagonal matrix.

All of the other answers are false.

The product of two diagonal matrices (in either order) is always another diagonal matrix.

Explanation:

You can verify this directly by proving it, or by multiplying a few examples on your calculator.

### Example Question #81 : Linear Algebra

Which of the following is true concerning the  identity matrix  ?

All of the other answers are true.

Explanation:

is the trace operation. It means to add up the entries along the main diagonal of the matrix. Since  has  ones along its main diagonal, the trace of  is .

### Example Question #1 : Operations And Properties

If

Find .

Explanation:

Since  is a diagonal matrix, we can find it's powers more easily by raising the numbers inside it to the power in question.

### Example Question #1 : The Identity Matrix And Diagonal Matrices

True or false, the set of all  diagonal matrices forms a subspace of the vector space of all  matrices.

False

True

True

Explanation:

To see why it's true, we have to check the two axioms for a subspace.

1. Closure under vector addition: is the sum of two diagonal matrices another diagonal matrix? Yes it is, only the diagonal entries are going to change, if at all. Nonetheless, it's still a diagonal matrix since all the other entries in the matrix are .

2. Closure under scalar multiplication: is a scalar times a diagonal matrix another diagonal matrix? Yes it is. If you multiply any number to a diagonal matrix, only the diagonal entries will change. All the other entries will still be .

### Example Question #1 : The Identity Matrix And Diagonal Matrices

True or false, if any of the main diagonal entries of a diagonal matrix is , then that matrix is not invertible.

False

True

True

Explanation:

Probably the simplest way to see this is true is to take the determinant of the diagonal matrix. We can take the determinant of a diagonal matrix by simply multiplying all of the entries along its main diagonal. Since one of these entries is , then the determinant is , and hence the matrix is not invertible.

### Example Question #2 : The Identity Matrix And Diagonal Matrices

True or False, the  identity matrix has  distinct (different) eigenvalues.

False

True

False

Explanation:

We can find the eigenvalues of the identity matrix by finding all values of  such that .

Hence we have

So  is the only eigenvalue, regardless of the size of the identity matrix.

### Example Question #83 : Linear Algebra

What is the name for a matrix obtained by performing a single elementary row operation on the identity matrix?

An elementary row matrix

A transition matrix

An elementary matrix

An inverse matrix

An elementary matrix

Explanation:

This is the correct term. Elementary matrices themselves can be used in place of elementary row operations when row reducing other matrices when convenient.

### Example Question #3 : The Identity Matrix And Diagonal Matrices

By definition, a square matrix that is similar to a diagonal matrix is

diagonalizable

the identity matrix

idempotent

symmetric

diagonalizable

Explanation:

Another way to state this definition is that a square matrix  is said to diagonalizable if and only if there exists some invertible matrix  and diagonal matrix  such that   .

### Example Question #4 : The Identity Matrix And Diagonal Matrices

The  identity matrix

has nullity .

is not diagonalizable.

is idempotent.

has rank .

has  distinct eigenvalues, regardless of size.