# Linear Algebra : The Identity Matrix and Diagonal Matrices

## Example Questions

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

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

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

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

All of the other answers are false.

The determinant of any diagonal matrix is .

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 #3 : The Identity Matrix And Diagonal Matrices

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 #4 : The Identity Matrix And Diagonal Matrices

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 #2 : 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 #6 : 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 #3 : 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 #8 : The Identity Matrix And Diagonal Matrices

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

An elementary matrix

An elementary row matrix

A transition 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 #9 : The Identity Matrix And Diagonal Matrices

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

the identity matrix

idempotent

diagonalizable

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 #10 : The Identity Matrix And Diagonal Matrices

The  identity matrix

is idempotent.

is not diagonalizable.

has nullity .

has rank .

has  distinct eigenvalues, regardless of size.

is idempotent.

Explanation:

An idempotent matrix is one such that . This is satisfied by the identity matrix since the identity matrix times itself is once again the identity matrix.

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