# 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 #2 : The Identity Matrix And Diagonal Matrices

Which of the following is true concerning diagonal matrices?

All of the other answers are false.

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 determinant of any diagonal matrix is .

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.

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 #5 : 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.

True

False

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 #7 : The Identity Matrix And Diagonal Matrices

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

True

False

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?

A transition matrix

An elementary matrix

An elementary row 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

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 #10 : The Identity Matrix And Diagonal Matrices

The  identity matrix

has nullity .

is idempotent.

has  distinct eigenvalues, regardless of size.

has rank .

is not diagonalizable.